A unied view on skewed distributions arising from selections

The Canadian Journal of Statistics 581Vol 34 No 4 2006 Pages 581ndash601La revue canadienne de statistique

A unified view on skewed distributionsarising from selectionsReinaldo B ARELLANO-VALLE Marcia D BRANCO and Marc G GENTON

Key words and phrasesKurtosis multimodal distribution multivariate distribution nonnormal distribu-tion selection mechanism skew-symmetric distribution skewness transformation

MSC 2000Primary 60E05 secondary 62E10

AbstractParametric families of multivariate nonnormal distributions have received considerable attentionin the past few decades The authors propose a new definition of a selection distribution that encompassesmany existing families of multivariate skewed distributions Their work is motivated by examples thatinvolve various forms of selection mechanisms and lead to skewed distributions They give the main prop-erties of selection distributions and show how various families of multivariate skewed distributions suchas the skew-normal and skew-elliptical distributions arise as special cases The authors further introduceseveral methods of constructing selection distributions based on linear and nonlinear selection mechanisms

Une perspective integree des lois asymetriques issues de processus de selectionResume Les familles parametriques de lois multivariees non gaussiennes ont suscite beaucoup drsquointeretdepuis quelques decennies Les auteurs proposent une nouvelle definition du concept de loi de selectionqui englobe plusieurs familles connues de lois asymetriques multivariees Leurs travaux sont motives pardiverses situations faisant intervenir des mecanismes de selection et conduisanta des lois asymetriquesIls mentionnent les principales proprietes des lois de selection et montrent comment diverses familles delois asymetriques multivariees telles que les lois asymetriques normales ou elliptiquesemergent comme casparticuliers Les auteurs presentent en outre plusieurs methodes de construction de lois de selection fondeessur des mecanismes lineaires ou non lineaires

1 INTRODUCTION

Since the collection and modelling of data plays a fundamental role in scientific research mul-tivariate distributions are central to statistical analyses Indeed multivariate distributions serveto model dependent outcomes of random experiments Despite the key role of the classical mul-tivariate normal distribution in statistics there has been a sustained interest among statisticiansin constructing nonnormal distributions In recent years significant progress has been madetoward the construction of so-called multivariate skew-symmetric and skew-elliptical distribu-tions and their successful application to problems in areas such as engineering environmetricseconomics and the biomedical sciences see the book edited by Genton (2004a) for a first andunique synthesis of results and applications in this topic

These multivariate skew-elliptical distributions and other related versions are essential forstatistical modelling for two main reasons Firstly they provide flexible parametric classes ofmultivariate distributions for the statisticianrsquos toolkit that allow for modelling key characteristicssuch as skewness heavy tails and in some cases surprisingly multimodality They also containthe multivariate normal distribution as a special case enjoy pleasant theoretical properties anddo not involve the often difficult task of multivariate data transformation Secondly they appearin the natural and important context of selection models that is when latent variables influencethe data collected We start by discussing these aspects in the next two sections

11 Parametric multivariate nonnormal distributions

In the last few decades applications from environmental financial and the biomedical sciencesamong other fields have shown that data sets following a normal law are more often the ex-ception rather than the rule Therefore there has been a growing interest in the construction of

582 ARELLANO-VALLE BRANCO amp GENTON Vol 34 No 4

parametric classes of nonnormal distributions particularly in models that can account for skew-ness and kurtosis A popular approach to model departure from normality consists of modifyinga symmetric (normal or heavy tailed) probability density function of a random variablevectorin a multiplicative fashion thereby introducing skewness This idea has been in the literaturefor a long time (eg Birnbaum 1950 Nelson 1964 Weinstein 1964 Roberts 1966 OrsquoHagan ampLeonard 1976) But it was Azzalini (1985 1986) who thoroughly implemented this idea forthe univariate normal distribution yielding the so-called skew-normal distribution An exten-sion to the multivariate case was then introduced by Azzalini amp Dalla Valle (1996) Statisticalapplications of the multivariate skew-normal distribution were presented by Azzalini amp Capi-tanio (1999) who also briefly discussed an extension to elliptical densities Arnold Castillo ampSarabia (1999 2001) discussed conditionally specified distributions related to the above con-struction for skewness

Since then many authors have tried to generalize these developments to skewing arbitrarysymmetric probability density functions with very general forms of multiplicative functionsSome of those proposals even allow for the construction of multimodal distributions In thepresent article we will provide an overview of existing distributions and novel extensions from aunified point of view

Besides the appeal of extending the classical multivariate normal theory the current impetusin this field of research is its potential for new applications For example a skewed version oftheGARCHmodel (generalized autoregressive conditional heteroscedasticity) in economics therepresentation of the shape of a human brain with skewed distributions in image analysis or askewed Kalman filter for the analysis of climatic time series are just but a few examples of thesemany applications see Genton (2004a) for those and more

12 Selection mechanisms

In addition to the attractive features of the above distributions for parametric modelling of mul-tivariate nonnormal distributions (a flexible parametric alternative to multivariate nonparametricdensity estimation) there is also a natural and important context where these distributions appearnamely selection models (see Heckman 1979 and more recently Copas amp Li 1997) Indeedconsider the model where a random vectorX isin IRp is distributed with symmetric probabilitydensity functionf(xθ) whereθ is a vector of unknown parameters The usual statistical analy-sis assumes that a random sampleX1 Xn from f(xθ) can be observed in order to makeinference aboutθ Nevertheless there are many situations where such a random sample mightnot be available for instance if it is too difficult or too costly to obtain If the probability den-sity function is distorted by some multiplicative nonnegative weight functionw(xθη) whereη denotes some vector of additional unknown parameters then the observed data is a randomsample from a weighted distribution (see eg Rao 1985) with probability density function

f(xθ)w(xθη)

Ew(Xθη) (1)

In particular if the observed data are obtained only from a selected portion of the populationof interest then (1) is called a selection model For example this can happen if the vectorX ofcharacteristics of a certain population is measured only for individuals who manifest a certaindisease due to cost or ethical reasons see the survey article by Bayarri amp DeGroot (1992) andreferences therein If the weight function satisfiesEw(Xθη) = 12 the resulting proba-bility density function (1) has a normalizing constant that does not depend on the parametersθ

andη A weight function with such a property can naturally occur when the selection criterionis such that a certain variable of interest is larger than its expected value given other variables

A simple example of selection is given by dependent maxima Consider two random vari-ables with bivariate normal distribution mean zero variance one and correlationρ The distri-bution of the maximum of those two random variables is exactly skew-normal (Roberts 1966

2006 SKEWED DISTRIBUTIONS 583

Loperfido 2002) with skewness depending onρ Note that whenρ = 0 the distribution of themaximum is still skewed For instance imagine students who report only their highest gradebetween calculus and geometry say The resulting distribution will be skewed due to this se-lective reporting although the joint bivariate distribution of the two grades might be normal orelliptically contoured The distribution of the minimum of the two random variables above isalso skew-normal but with a negative sign for the skewness parameter

Another example of selection arises in situations where observations obey a rather well-behaved law (such as multivariate normal) but have been truncated with respect to some hid-den covariables see eg Arnold amp Beaver (2000a 2002 2004) Capitanio Azzalini ampStanghellini (2003) A simple univariate illustration is the distribution of waist sizes for uni-forms of airplane crews who are selected only if they meet a specific minimal height require-ment A bivariate normal distribution might well be acceptable to model the joint distributionof height and waist measurements However imposition of the height restriction will result ina positively skewed distribution for the waist sizes of the selected individuals Note that thistype of truncation might occur without our knowing of its occurrence In that case it is called ahidden truncation Of course selection mechanisms are not restricted to lower truncations butcan also involve upper truncations or various more complex selection conditions For instanceairplane crews cannot be too tall because of the restrictive dimensions of the cabin So the heightrequirements for airplane crews effectively involve both lower and upper truncations

Astronomy is another setting where hidden truncations occur The impact of observationalselection effects known generically in the literature as Malmquist bias (Malmquist 1920) hasbeen at the centre of much controversy in the last decades This effect occurs when an astronomerobserves objects at a distance Indeed instruments set a limit on the faintest objects (such asstars for example) that we can see In other words in any observation there is a minimum fluxdensity (measure of how bright objects seem to us from Earth) below which we will not detectan object This flux density is proportional to the luminosity (measure of how bright objects areintrinsically) divided by the square of the distance so it is possible to see more luminous objectsout to larger distances than intrinsically faint objects This means that the relative numbers ofintrinsically bright and faint objects that we see may be nothing like the relative numbers per unitvolume of space Instead bright objects are over-represented and the average luminosity of theobjects we see inevitably increases with distance There is not really anything that can be doneabout the Malmquist bias at this point in our stage of technology and observation except merelyto correct for it in the statistical analysis Neglecting this selection bias will result in skewedconclusions

13 Objectives and summary

The main objective of this article is to develop a unified framework for distributions arising fromselections and to establish links with existing definitions of multivariate skewed distributionsBecause many definitions have emerged in recent years we believe that it is necessary to providea crisp overview of this topic from a unified perspective We hope that this account will serve asa resource of information and future research topics to experts in the field new researchers andpractitioners

Statistical inference for those families of multivariate skewed distributions is still mostly un-resolved and problems have been reported already in the simplest case of the skew-normal dis-tribution It is currently the subject of vigorous research efforts and recent advances on this topicinclude work by Azzalini (2005) Azzalini amp Capitanio (1999 2003) DiCiccio amp Monti (2004)Ma Genton amp Tsiatis (2005) Ma amp Hart (2006) Pewsey (2000 2006) and Sartori (2006) Itis hence beyond the scope of this article to review inferential aspects Future research on in-ferential issues for skewed distributions arising from selections both from a frequentist and aBayesian perspective will however lead to the ultimate goal of facilitating the development ofnew applications

The structure of the present article is the following In Section 2 a definition of a selection


distribution that unifies many existing definitions of multivariate skewed distribution availablein the literature is proposed The main properties of selection distributions are also derived andtherefore unify the properties of these existing definitions In Section 3 we describe in detailthe link between selection distributions and multivariate skewed distributions developed in theliterature such as skew-symmetric and skew-elliptical distributions among many others InSection 4 several methods for constructing selection distributions based on linear and nonlinearselection mechanisms are introduced A discussion is provided in Section 5

2 SELECTION DISTRIBUTIONS

We will begin by proposing a definition of a multivariate selection distribution and discuss someof its main properties

DEFINITION 1 LetU isin IRq andV isin IRp be two random vectors and denote byC a measurablesubset of IRq We define a selection distribution as the conditional distribution ofV givenU isin Cthat is as the distribution of(V |U isin C) We say that a random vectorX isin IRp has a selection

distribution ifXd= (V |U isin C) We use the notationX sim SLCTpq with parameters depending

on the characteristics ofU V andCThis definition is simply a conditional distribution but it serves to unify existing definitions

of skewed distribution arising from selection mechanisms as well as to provide new classesof multivariate symmetric and skewed distributions Of course ifC = IRq then there is noselection and so we assume that0 lt P(U isin C) lt 1 in the sequel IfU = V then the selectiondistribution is simply a truncated distribution described byV givenV isin C Moreover ifV =(UT) then a selection distribution corresponds to the joint distribution ofU andT (marginally)truncated onU isin C from which we can obtain by marginalization both the truncated distributionof (U |U isin C) and the selection distribution of(T |U isin C)

21 Main properties

The main properties of a selection random vectorXd= (V |U isin C) can be studied easily by

examining the properties of the random vectorsV andU Indeed ifV in Definition 1 has aprobability density function (pdf)fV say thenX has a pdffX given by

fX(x) = fV(x)P(U isin C |V = x)

P(U isin C) (2)

See for example Arellano-Valle and del Pino (2004) for this and see Arellano-Valle delPino amp San Martın (2002) for the particular choice ofC = u isin IRq |u gt 0 where theinequality between vectors is meant componentwise As motivated by the selection mecha-nisms in Section 12 our interest lies in situations where the pdffV is symmetric In thatcase the resulting pdf (2) is generally skewed unless the conditional probabilities satisfyP(U isin C |V = x) = P(U isin C |V = minusx) for all x isin IRp For instance this is the caseif U andV are independent but this is not a necessary condition

Further properties of the selection random vectorXd= (V |U isin C) can be studied directly

from its definition For instance we have

(P1) When(UV) has a joint densityfUV say an alternative expression for the pdf ofXd=

(V |U isin C) is given by

fX(x) =

intC

fUV(ux) duintC

fU(u) du (3)

wherefU denotes the marginal pdf ofU Expression (3) is useful to compute the cumula-tive distribution function (cdf) as well as the moment generating function (mgf) ofX


(P2) For any Borel functiong we haveg(X)d= (g(V) |U isin C) Moreover since

(g(V) |U isin C) is determined by the transformation(UV) rarr (U g(V)) then the pdfof g(X) can be computed by replacingX with g(X) andV with g(V) in (2) (or (3))respectively If a family of distributions ofV is closed under a set of transformationsg(for instance all linear transformations) then the corresponding family of selection distri-butions is also closed under this set of transformations If a family of distributions ofV

is closed under marginalization (for instance ifV is multivariate normal then subvectorsof V are multivariate normal) then the corresponding family of selection distributions isalso closed under marginalization

(P3) The conditional selection distribution of(X2 |X1 = x1) where the joint distribution of

(X1X2)d= [(V1V2) |U isin C] can be obtained by noting that it is equivalent to consider

the conditional selection distribution of(V2 |V1 = x1U isin C) By (2) it has a pdf givenby

fX2 |X1=x1(x2) = fV2 |V1=x1

(x2)P(U isin C |V1 = x1V2 = x2)

P(U isin C |V1 = x1) (4)

Consider also the partitionU = (U1U2) such that(U1V1) and (U2V2) are in-dependent and assume thatC = C1 times C2 is such thatU isin C is equivalent toU1 isin C1 andU2 isin C2 Then under these conditions it follows directly from (4) that

X1d= (V1 |U isin C)

d= (V1 |U1 isin C1) andX2

d= (V2 |U isin C)

d= (V2 |U2 isin C2) and

that they are independent For more details on these properties but considering the partic-ular subsetC = u isin IRq |u gt 0 see Arellano-Valle amp Genton (2005) and Arellano-Valle amp Azzalini (2006)

Selection distributions depend also on the subsetC of IRq One of the most important selec-tion subset is defined by

C(β) = u isin IRq |u gt β (5)

whereβ is a vector of truncation levels In particular the subsetC(0) leads to simple selectiondistributions Note however that the difference betweenC(β) andC(0) is essentially a changeof location Another interesting subset consists of upper and lower truncations and is defined by

C(αβ) = u isin IRq |α gt u gt β (6)

see the example in Section 12 about height requirements for airplane crews with both lower andupper truncations Arnold Beaver Groeneveld amp Meeker (1993) considered the case ofC(αβ)with p = q = 1 Many other subsets can be constructed for instance quadratic-based subsetsof the formu isin IRq |u⊤Qu gt β whereQ is a matrix andβ isin IR a truncation level seeGenton (2005) for more examples

22 Selection elliptical distributions

Further specific properties of selection distributions can be examined when we know the jointdistribution ofU andV A quite popular family of selection distributions arises whenU andV

have a joint multivariate elliptically contoured (EC) distribution that is(

U

V

)sim ECq+p

(ξ =

(ξU

ξV

)Ω =

(ΩU ∆⊤

∆ ΩV

) h(q+p)

) (7)

whereξU

isin IRq andξV

isin IRp are location vectorsΩU isin IRqtimesq ΩV isin IRptimesp and∆ isin IRptimesq

are dispersion matrices and in addition to these parametersh(q+p) is a density generator func-tion We denote the selection distribution resulting from (7) bySLCT -ECpq(ξΩ h(q+p) C)They typically result in skew-elliptical distributions except for the cases (a)∆ = O a matrix


of zeros andh(q+p) is the normal density generator function or (b)∆ = O andC = C(ξU

) de-fined according to (5) since in both cases we have in (2) thatP(U isin C |V = x) = P(U isin C)for all x In fact conditions in (a) imply thatU andV are independent while conditions in (b)imply thatP(U isin C |V = x) = P(U isin C) = Φq(00ΩU) for all x and any density generatorfunction h(q+p) whereΦq( middot 0ΩU) denotes the cdf of theNq(0ΩU) distribution In otherwords the condition thatU andV be correlated random vectors ie∆ 6= O is important inorder to avoid in (2) that the skewing factorP(U isin C |V = x)P(U isin C) be equal to one forall x see also Section 4

In general any selection distribution can be constructed hierarchically by specifying a mar-ginal distribution forV and then a conditional distribution for(U |V = x) For example thedistributionSLCT -ECqp(ξΩ h(q+p) C) that follows from(7) according to Definition 1 canbe obtained by specifying

V sim ECp(ξVΩV h(p)) and (8)

(U |V = x) sim ECq(ξU+ ∆⊤Ωminus1

V(x minus ξ

V)ΩU minus ∆⊤Ωminus1

V∆ h

(q)υ(x))

whereh(q)υ(x)(u) = h(q+p)(u + υ(x))h(p)(υ(x)) is the induced conditional generator and

υ(x) = (xminusξV

)⊤Ωminus1V

(xminusξV

) It is then straightforward to obtain the pdf ofXd= (V |U isin C)

by means of (2) In fact let

fr(y ξY

ΩY h(r)) = |ΩY|minus12h(r)((y minus ξY

)⊤Ωminus1Y

(y minus ξY

))

be the pdf of ar-dimensional elliptically contoured random vectorY sim ECr(ξYΩY h(r))

and denoteP(Y isin C) by F r(C ξY

ΩY h(r)) ie

F r(C ξY

ΩY h(r)) =

int

C

fr(y ξY

ΩY h(r)) dy

Then applying (8) in (2) we have the following generic expression for the pdf ofXd=

(V |U isin C)

fX(x) = fp(x ξV

ΩV h(p))F q(C∆⊤Ωminus1

V(x minus ξ

V) + ξ

UΩU minus ∆⊤Ωminus1

V∆ h

(q)υ(x))

F q(C ξU

ΩU h(q)) (9)

which can also be easily rewritten according to (3) as

fX(x) =

intC

fq+p(ux ξΩ h(q+p)) duintC

fq(u ξU

ΩU h(q)) du (10)

Expression (10) can be helpful to compute the cdf and also the mgf of any selection ellipticalrandom vectorX sim SLCT -ECqp(ξΩ h(q+p) C)

Note finally that depending on the choice ofh(q+p) (9) (or (10)) becomes the pdf of differentselection elliptical distributions For example whenh(q+p)(u) = c(q+p ν)1+uminus(q+p+ν)2wherec(r ν) = Γ[(r + ν)2]νν2(Γ[ν2]πr2) (9) becomes the selection-tpdf with ν degreesof freedom given by

fX(x) = tp(x ξV

ΩV ν)

T q(C∆⊤Ωminus1V

(x minus ξV

) + ξU

ν + υ(x)

ν + p[ΩU minus ∆⊤Ωminus1

V∆] ν + p)

T q(C ξU

ΩU ν)

(11)wheretr( middot ξΩ ν) denotes the pdf of a Student-trandom vectorY sim Studentr(ξΩ ν) ofdimensionr andT r(C ξΩ ν) = P(Y isin C) This produces a heavy tailed selection distri-bution that is useful for robustness purposes Due to its importance the normal case for whichh(r)(u) = φr

(radicu)

= (2π)minusr2 expminusu2 u ge 0 is considered next in a separate section


23 The selection normal distribution

Within the elliptically contoured class (7) one of the most important cases is undoubtedly whenU andV are jointly multivariate normal so that instead of (7) we have

(U

V

)sim Nq+p

(ξ =

(ξU

ξV

)Ω =

(ΩU ∆⊤

∆ ΩV

)) (12)

We denote the selection distribution resulting from (12) bySLCT -Npq(ξΩ C) We estab-lish two important theoretical characteristics of theSLCT -Npq(ξΩ C) distributions namelytheir pdf and their mgf for any selection subsetC isin IRq This result unifies the moment prop-erties of existing definitions of skewed distributions based on the multivariate normal distribu-tion Specifically letX have a multivariate selection distribution according to Definition 1 withU andV jointly normal as in (12) that isX sim SLCT -Npq(ξΩ C) Let φr( middot ξY

ΩY)be the pdf of anr-dimensional normal random vectorY sim Nr(ξY

ΩY) and denote byΦr(C ξ

YΩY) = P(Y isin C) It is then straightforward to derive the conditional distribution in

Definition 1 yielding the following pdf based on (2)

fX(x) = φp(x ξV

ΩV)Φq(C ∆⊤Ωminus1

V(x minus ξ

V) + ξ


V∆)

Φq(C ξU

ΩU) (13)

which obviously follows also from (9) whenh(q+p) is chosen as the normal density generator Inaddition the moment generating function ofX is

MX(s) = exp

s⊤ξ

V+

1

2s⊤ΩVs

Φq(C∆⊤s + ξ

UΩU)

Φq(C ξU

ΩU) (14)

The proof of this result is provided in the Appendix From (13) and (14) we can also derive thefollowing stochastic representation (or transformation) for the normal selection random vector

Xd= (V |U isin C) ie withU andV jointly normal as in (12) LetX0 sim Nq(0ΩU) andX1 sim

Np(0ΩV minus ∆Ωminus1U

∆⊤) be independent normal random vectors Note by (12) that(UV)d=

(ξU

+ X0 ξV+ ∆Ωminus1

UX0 + X1) Thus considering also a random vectorX0[C minus ξ

U]

d=

(X0 |X0 isin C minus ξU

) which is independent ofX1 we have

Xd= ξ

V+ ∆Ωminus1

UX0[C minus ξ

U] + X1 (15)

All these results reduce to similar ones obtained in Arellano-Valle amp Azzalini (2006) whenC =C(0) The stochastic representation (15) is useful for simulations from selection distributionsand for computations of their moments For instance the mean vector and covariance matrix ofX are given by

E(X) = ξV

+ ∆Ωminus1U

E(X0[C minus ξU

])

var(X) = ΩV + ∆Ωminus1U

ΩU minus var(X0[C minus ξU

])Ωminus1U

∆⊤

respectively Note that in general it is not easy to obtain closed-form expressions for the ldquotrun-catedrdquo mean vectorE(X0[C minus ξ

U]) and covariance matrixvar(X0[C minus ξ

U]) for any selection

subsetC However for the special case whereΩU is diagonal andC = C(αβ) the com-ponents of(X0 |X0 isin C) are independent Therefore the computation of these moments isequivalent to computing marginal truncated means and variances of the formE(X | a lt X lt b)andvar(X | a lt X lt b) whereX sim N(micro σ2) for which the results of Johnson Kotz amp Balakr-ishnan (1994sect 101) can be used see also Arellano-Valle amp Genton (2005) and Arellano-Valle ampAzzalini (2006) for more details on similar results


24 Scale mixture of the selection normal distribution

Another important particular case that follows from the elliptically contoured class (7) is obtainedwhen the joint distribution ofU andV is such that

(U

V

) ∣∣∣W = w sim Nq+p

(ξ =

(ξU

ξV

) wΩ =

(wΩU w∆⊤

w∆ wΩV

)) (16)

for some nonnegative random variableW with cdf G This is equivalent to considering in (7)a density generator function such thath(q+p)(u) =

intinfin

0(2πw)minus(q+p)2 expminusu2w dG(w)

For example ifG is the cdf of the inverted gamma distribution with parametersν2 andν2IG(ν2 ν2) say thenh(q+p)(u) = c(q + p ν)1 + uminus(q+p+ν)2 wherec(r ν) is defined inSection 22 yielding the selection-tdistribution withν degrees of freedom and pdf (11)

The properties of any selection random vectorXd= (V |U isin C) that follows from (16) can

be studied by noting that(X |W = w) sim SLCT -Npq(ξ wΩ C) whereW sim G In particularfor the pdf ofX we havefX(x) =

intinfin

0fX|W=w(x) dG(w) wherefX |W=w(x) is the selection

normal pdf (13) withΩ replaced bywΩ ie

fX(x) =

int infin

0

φp(x ξV

wΩV)Φq(C ∆⊤Ωminus1

V(x minus ξ

V) + ξ

U wΩU minus ∆⊤Ωminus1

V∆)

Φq(C ξU

wΩU)dG(w)

Alternatively from (3) or (10) it follows that the pdf ofX can also be computed as

fX(x) =

intC

intinfin

0φq+p(ux ξ wΩ) dG(w) duint

C

intinfin

0φq(u ξ

U wΩU) dG(w) du

Similar results for the mgf can be obtained by considering (14) Note however that the mean andcovariance matrix ofX can be computed (provided they exist) by using the well-known facts thatE(X) = E[E(X |W )] andvar(X) = var[E(X |W )] + E[var(X |W )] whereE(X |W = w)andvar(X |W = w) are the mean and covariance matrix corresponding to the (conditional)selection normal random vector(X |W = w) sim SLCT -Npq(ξ wΩ C) see Section 23 Ad-ditional properties of this subclass of selection elliptical models can be explored for particularchoices ofC For instance whenC = C(ξ

U) it is easy to show from (15) that

Xd= ξ

V+

radicW XN (17)

whereW sim G andXN sim SLCT -Npq(0Ω C(ξU

)) and they are independent In particularif W has finite first moment then (17) yieldsE(X) = ξ

V+ E

(radicW)E(XN ) andvar(X) =

E(W ) var(XN ) More generally sinceΦq(C(β) ξY

ΩY) = Φq(ξYminus β0ΩY) the cdf of a

Nq(0ΩY) distribution evaluated atξYminus β then from (17) and (14) we have for the mgf ofX

that

MX(s) = E[MV |W (s)]Φq(∆

⊤s0ΩU)

Φq(00ΩU) (18)

whereMV |W (s) = exps⊤ξ

V+ 1

2ws⊤ΩVs

since(V |W = w) sim Np(ξV wΩV) and

W sim G The proof of this result is provided in the Appendix From (18) it is clear that themoment computations reduce substantially when the matrixΩU is diagonal The special caseof the multivariate skew-tdistribution that follows whenG is theIG(ν2 ν2) distribution isstudied in Arellano-Valle amp Azzalini (2006)

3 LINK WITH SKEWED MULTIVARIATE DISTRIBUTIONS

We show how many existing definitions of skew-normal skew-elliptical and other related dis-tributions from the literature can be viewed as selection distributions in terms of assumptions


on the joint distribution of(UV) and onC We start from the most general class of funda-mental skewed distributions (Arellano-Valle amp Genton 2005) and by successive specializationsteps work down to the very specific skew-normal distribution of Azzalini (1985) We describethe links between those various distributions and provide a schematic illustrative summary ofselection distributions and other skewed distributions

31 Fundamental skewed distributions

Fundamental skewed distributions have been introduced by Arellano-Valle amp Genton (2005)They are selection distributions with the particular selection subsetC(0) defined in (5) andtherefore the pdf (2) becomes

fX(x) = fV(x)P(U gt 0 |V = x)

P(U gt 0) (19)

When the pdffV is symmetric or elliptically contoured the distributions resulting from (19) arecalled fundamental skew-symmetric(FUSS) and fundamental skew-elliptical(FUSE) respec-tively If fV is the multivariate normal pdf then the distribution resulting from (19) is calledfundamental skew-normal(FUSN) In this case note that the joint distribution ofU andV is notdirectly specified and in particular does not need to be normal The only requirement is to havea model forP(U gt 0 |V = x) Such an example is provided in Section 35 Finally ifU andV are jointly multivariate normal as in (12) then (19) takes the form of (13) withC = C(0)and Φr( middot middot middot ) = Φr( middot ξΩ) the cdf of aNr(ξΩ) distribution In particular ifξ

U= 0

ξV

= 0 ΩU = Iq andΩV = Ip then the resulting distribution is calledcanonical fundamentalskew-normal(CFUSN) with the pdf reducing to the form

fX(x) = 2qφp(x)Φq(∆⊤x0 Iq minus ∆⊤∆) (20)

Location and scale can be further reincorporated in (20) by considering the pdf ofmicro + Σ12Xwheremicro isin IRp is a location vectorΣ isin IRptimesp is a positive definite scale matrix This family ofdistributions has many interesting properties such as a stochastic representation a simple formulafor its cdf characteristics of linear transformations and quadratic forms among many otherssee Arellano-Valle amp Genton (2005) for a detailed account Fundamental skewed distributionsappear also naturally in the context of linear combinations of order statistics see Crocetta ampLoperfido (2005) and Arellano-Valle amp Genton (2006ab)

32 Unified skewed distributions

Unified skewed distributions have been introduced by Arellano-Valle amp Azzalini (2006) TheyareFUSNor FUSEdistributions with the additional assumption that the joint distribution ofU

andV is multivariate normal or elliptically contoured as in (12) or (7) respectively These uni-fied skewed distribution are calledunified skew-normal(SUN) andunified skew-elliptical(SUE)respectively and they unify (contain or are equivalent under suitable reparameterizations to) allthe skew-normal or skew-elliptical distributions derived by assuming (12) or (7) respectively Infact with some modification of the parameterization used by Arellano-Valle amp Azzalini (2006)theSUEpdf is just the selection pdf that follows from (7) withC = C(0) and has the form

fX(x) = fp(x ξV

ΩV h(p))Fq(ξU

+ ∆⊤Ωminus1V

(x minus ξV

)0ΩU minus ∆⊤Ωminus1V

∆ h(q)υ(x))

Fq(ξU0ΩU h(q))

(21)

where (as before)fp(x ξV

ΩV h(p)) = |ΩV|minus12h(p)(υ(x)) υ(x) = (xminusξV

)⊤Ωminus1V

(xminusξV

)

and h(q)υ(x)(u) = h(p+q)(u + υ(x))h(p)(υ(x)) and Fr(z0Θ g(r)) denote the cdf of the

ECr(0Θ g(r)) distribution namelyFr(z0Θ g(r)) =intvlez

|Θ|minus12g(r)(v⊤Θminus1v) dv The

SUN pdf follows by taking the normal generatorh(p+q)(u) = (2π)minus(p+q)2eminusu2 u ge 0


for which h(r)υ (u) = h(r)(u) = (2π)minusr2eminusu2 u υ ge 0 and so (21) takes the form of

(13) with C = C(0) and Φr( middot middot middot ) = Φr( middot ξΩ) The multivariate skew-t(ST) distri-bution considered in Branco amp Dey (2001) and Azzalini amp Capitanio (2003) is extended inArellano-Valle amp Azzalini (2006) by considering in (21) thet-generator withν degrees offreedom defined byh(p+q)(u) = c(p + q ν)ν + uminus(p+q+ν)2 u ge 0 for which h(p) =

c(p ν)ν + uminus(p+ν)2 andh(q)υ (u) = c(q ν + p)ν + υ + uminus(p+q+ν)2 u υ ge 0 where

c(r ω) = Γ[(r + ω)2]ωω2Γ[ω2]πr2 see also Section 24

33 Closed skewed distributions

The closed skew-normal(CSN) distributions introduced by Gonzalez-Farıas Domınguez-Molina amp Gupta (2004a) is alsoFUSN by reparameterization of the covariance matrix of thenormal joint distribution in (12) as∆ = ΩVD⊤ andΩU = Θ + DΩVD⊤ Another closedskew-normal distribution which was introduced in Arellano-Valle amp Azzalini (2006) followsalso as aFUSNwhen∆ = AΩU andΩV = Ψ + AΩUA⊤ in (12) An important property ofthis last parameterization is that it simplifies the stochastic representation in (15) to a simpler

structure of the formXd= ξ

V+ AX0[minusξ

U] + X1 whereX0[minusξ

U]

d= (X0 |X0 gt minusξ

U) with

X0 sim Nq(0ΩU) and is independent ofX1 sim Np(0Ψ) thus generalizing the skew-normaldistribution considered by Sahu Dey amp Branco (2003) As was shown in Arellano-Valle amp Az-zalini (2006) all these distributions are equivalent to theSUNdistribution under some suitablereparameterizations The termclosedwas motivated by the fact thatCSNdistributions are closedunder marginalizations and also under conditioning whenP(U isin C) 6= 2minusq Moreover theCSNdistributions have some particular additive properties see Gonzalez-Farıas Domınguez-Molina amp Gupta (2004b)

34 Hierarchical skewed distributions

Liseo amp Loperfido (2006) generalized a pioneer result obtained by OrsquoHagan amp Leonard (1976)from a Bayesian point of view who used the selection idea to incorporate constraints on para-meters in the prior specifications This is just an interesting example of how Bayesian modellingcan be viewed through selection distributions In fact suppose thatV |micro sim Np(a + AmicroΘ)micro sim Nq(0Γ) where we know also thatmicro + c gt 0 This is equivalent to considering themultivariate normal model (see also Arellano-Valle amp Azzalini 2006)

(micro

V

)sim Nq+p

((a

0

)

(Γ ΓA⊤

AΓ Θ + AΓA⊤

)) with micro + c gt 0

Thus the selection model we are considering isXd= (V |micro + c gt 0) which by (19) has pdf

fX(x) = φp(xaΣ)Φq(c + ∆⊤Σminus1(x minus a)0Γ minus ∆⊤Σminus1∆)

Φq(c0Γ) (22)

where∆ = AΓ andΣ = Θ+AΓA⊤ A random vector with pdf (22) is said to have ahierarchicalskew-normal(HSN) distribution and is also in theSUNfamily (Arellano-Valle amp Azzalini 2006)

35 Shape mixture of skewed distributions

The idea ofshape mixture of skew-symmetric(SMSS) distributions is introduced by Arellano-Valle Genton Gomez amp Iglesias (2006) The starting point is a skewed multivariate distri-bution with pdf of the form2pfp(x)

prodpi=1 G(λixi) or 2fp(x)G(λ⊤

x) with a (prior) distrib-ution for the shape parameterλ = (λ1 λp)

⊤ The distributions resulting from integra-tion on λ are SMSS and they are in theFUSSfamily Whenfp = φp the resulting distri-bution is called ashape mixture of skew-normal(SMSN) distribution Thus as was shown inArellano-Valle Genton Gomez amp Iglesias (2006) a particularSMSNrandom vectorX is ob-tained by Bayesian modelling by considering thatXi |λ sim SN(λi) i = 1 p are inde-pendent whereSN(λ) denotes the standard univariate skew-normal distribution introduced by


Azzalini (1985) see also Section 38 and adopting the priorλ sim Np(microΣ) This is equiv-

alent to considering the selection vectorXd= (V |U isin C(0)) with V sim Np(0 Ip) and

U |V = v sim Np(D(micro)v Ip + D(v)ΣD(v)) whereD(micro) is thep times p diagonal matrix withthe components of the vectormicro as the diagonal elements This example illustrates aFUSNcasewhere the marginal distribution ofV and the conditional distribution ofU given V = v arenormal but the corresponding joint distribution is not normal The case withp = 1 is discussedin Arellano-Valle Gomez amp Quintana (2004b) which is called skew-generalized normal (SGN)distribution and has a pdf given by

fX(x) = 2φ(x)Φ

(λ1xradic

1 + λ2x2

)

whereλ1 isin IR andλ2 ge 0 Note that this family of pdfrsquos contains both theN(0 1) pdf whenλ1 = 0 and theSN(λ1) whenλ2 = 0 From a shape mixture point of view it can be obtained byassuming thatX |λ sim SN(λ) with λ sim N(λ1 λ2) and from a selection approach it corresponds

to the pdf ofXd= (V |U gt 0) whereV sim N(0 1) andU |V = x sim N(λ1x 1 + λ2x

2)

36 Skew-symmetric and generalized skew-elliptical distributions

For all the skewed distributions considered in this and the next two sections we have the casewith dimensionq = 1 ie selection models considering constraints on one random variableU only Skew-symmetric (SS) distributions have been introduced by Wang Boyer amp Gen-ton (2004a) They are selection distributions with the particular selection subsetC(0) definedin (5) the dimensionq = 1 a symmetric random variableU and a symmetric random vec-tor V Therefore we haveP(U gt 0) = 12 and the functionπ(x) = P(U gt 0 |V = x)called askewing function must satisfy0 le π(x) le 1 andπ(minusx) = 1 minus π(x) The pdf of askew-symmetric distribution based on (2) then takes the form

fX(x) = 2fV(x)π(x) (23)

which is similar to the formulation in Azzalini amp Capitanio (2003) except for a different butequivalent representation ofπ(x)

Generalized skew-elliptical (GSE) distributions have been introduced by Genton amp Loper-fido (2005) They are skew-symmetric distributions with pdf given by (23) and with the addi-tional assumption that the distribution ofV is elliptically contoured When the pdffV is mul-tivariate normal or multivariatet the resulting distributions are calledgeneralized skew-normal(GSN) andgeneralized skew-t(GST) respectively

Both skew-symmetric and generalized skew-elliptical distributions have a simple stochas-tic representation and possess straightforward properties for their linear transformations andquadratic forms Genton (2004b) provides a detailed overview of skew-symmetric and gener-alized skew-elliptical distributions with an illustrative data fitting example

37 Flexible skewed distributions

Flexible skewed distributions have been introduced by Ma amp Genton (2004) They are skew-symmetric distributions with a particular skewing function Indeed any continuous skewingfunction can be written asπ(x) = H(w(x)) whereH IR rarr [0 1] is the cdf of a continuousrandom variable symmetric around zero andw IRp rarr IR is a continuous function satisfyingw(minusx) = minusw(x) By using an odd polynomialPK of orderK to approximate the functionwwe obtain a flexible skew-symmetric (FSS) distribution with pdf of the form

fX(x) = 2fV(x)H(PK(x)) (24)

Under regularity conditions on the pdffV Ma amp Genton (2004) have proved that the class ofFSSdistributions is dense in the class ofSSdistributions under theLinfin norm This means that


FSSdistributions can approximateSSdistributions arbitrarily well Moreover the number ofmodes in the pdf (24) increases with the dimensionK of the odd polynomialPK whereas onlyone mode is possible whenK = 1 When the pdffV is multivariate normal ort the resultingdistributions are calledflexible skew-normal(FSN) andflexible skew-t(FST) respectively Thusflexible skewed distributions can capture skewness heavy tails and multimodality

38 Skew-elliptical distributions

Skew-elliptical (SE) distributions have been introduced by Branco amp Dey (2001) as an extensionof the skew-normal distribution defined by Azzalini amp Dalla Valle (1996) to the elliptical classThey are selection distributions considering a random variableU (ie q = 1) and a randomvectorV which have a multivariate elliptically contoured joint distribution as given in (7) withξU

= 0 ΩU = 1 ie they are in theSUEsubclass with dimensionq = 1 Thus by (21) theSEpdf has the form

fX(x) = 2fp(x ξV

ΩV h(p))F1(α⊤(x minus ξ

V) h

(1)υ(x))

with α = (1 minus ∆⊤Ωminus1V

∆)minus12Ωminus1V

∆ andυ(x) andh(1)υ(x) defined in Section 32

The idea of anSEdistribution was also considered by Azzalini amp Capitanio (1999) who in-troduced a different family ofSEdistributions than the above class and suggested exploring theirproperties as a topic for future research The relationship between both families is discussedin Azzalini amp Capitanio (2003) who have shown that they are equivalent in many importantcases like the skew-t(ST) distribution studied by Branco amp Dey (2001) and Azzalini amp Capi-tanio (2003) and extended forq gt 1 in Arellano-Valle amp Azzalini (2006) as a member ofSUEdistributions TheSTpdf takes the form

fX(x) = 2tp(x ξV

ΩV ν)T1

((ν + 1

ν + υ(x)

)12

α⊤(x minus ξV

) ν + 1

)

wheretp(xθΘ ϑ) is the pdf of thep-dimensional Student-tdistribution with locationθ scaleΘ andϑ degrees of freedom andT1(xϑ) is the cdf of the standard univariate Student-tdistri-bution withϑ degrees of freedom Other versions of the skew-tdistribution were proposed byBranco amp Dey (2002) and Sahu Dey amp Branco (2003) Another important case is of course themultivariateSNp(ξV

ΩVα) distribution introduced by Azzalini amp Dalla Valle (1996) whosepdf has the form

fX(x) = 2φp(x ξV

ΩV)Φ(α⊤(x minus ξV

)) (25)

and is just a member of theSUNdistributions with dimensionq = 1 If in additionp = 1 then(25) reduces to the pdf of the seminal univariate skew-normal distribution of Azzalini (19851986) see also Gupta amp Chen (2004) for a slightly different definition of a multivariate skew-normal distribution In all of the above modelsα is a p-dimensional vector controlling theskewness andα = 0 recovers the symmetric parent distribution

Other particular skew-elliptical distributions include skew-Cauchy (Arnold amp Beaver 2000b)skew-logistic (Wahed amp Ali 2001) skew-exponential power (Azzalini 1986) and skew-slash(Wang amp Genton 2006)

A fundamental property of all the selection distributions satisfyingP(U isin C) = 12has been mentioned extensively in the literature see eg Azzalini (2005) Azzalini amp Capi-tanio (2003) Genton (2005) Genton amp Loperfido (2005) Wang Boyer amp Genton (2004b) forrecent accounts In brief the distribution of even functions of a random vector from such a dis-tribution does not depend onP(U isin C |V) ie the skewness This distributional invariance isof particular interest with respect to quadratic forms because of their extensive use in Statistics


SLCT

COther subsetsβ

FUSE

FSS

SUN

CFUSNCSN

SMSN

SESUEFUSN

SE

GSE

SN

GSNFSN FSE

SS

HSN

q=1

FUSS C( )

etc

C( minus )

C( ) C(0)

α

α

β

α

SMSS

FIGURE 1 Schematic illustrative summary of selection distributions

39 A brief summary of skewed distributions

The field of skew-normal and related distributions has been growing fast in recent years andmany new definitions of univariate and multivariate skewed distributions have been proposedWe believe that new researchers in this field (as well as experts) can be confused by the numberof different definitions and their links In addition to the unified overview described in the previ-ous sections we propose in Figure 1 a schematic illustrative summary of the main multivariateskewed distributions defined in the literature based on our selection distribution point of view

The left panel of Figure 1 describes selection distributions based on the selection subsetC(β)defined in (5) and essentially contains the definitions of univariate and multivariate skewed dis-tributions described previously Note that up to a location shift this is equivalent to usingC(0)The top part corresponds toq gt 1 conditioning variables whereas the bottom part is forq = 1The right panel of Figure 1 describes selection distributions based on arbitrary selection subsetsC for example such asC(αβ) defined in (6) orC(αminusα) Those selection distributionsare mainly unexplored except for some general properties derived in Section 2 In Table 1 weprovide also an alphabetical summary of the abbreviations of skewed distributions in order tocomplement Figure 1 as well as some of their main references Note that for the sake of clarityin Figure 1 we represent only skewed distributions based on normal elliptically contoured andsymmetric probability density functions Skewed distributions such as skew-tand skew-Cauchy


are implicitly included in the skew-elliptical families Similarly Table 1 is not meant to be ex-haustive since so many definitions have appeared in the literature but it should be useful to traceback some of the main references

TABLE 1 Alphabetical summary of selection distributions with main references

ABBREVIATION NAME OF DISTRIBUTION MAIN REFERENCES

CFUSN Canonical fundamental skew-normal Arellano-Valle amp Genton (2005)

CSN Closed skew-normal Gonzalez-Farıas et al (2004a)

Arellano-Valle amp Azzalini (2006)

FSN FSS FST Flexible skew -normal -symmetric -t Ma amp Genton (2004)

FUSEFUSN Fundamental skew -elliptical -normal Arellano-Valle amp Genton (2005)

FUSS -symmetric

GSE GSN GST Generalized skew -elliptical normal -t Genton amp Loperfido (2005)

HSN Hierarchical skew-normal Liseo amp Loperfido (2006)

SMSN SMSS Shape mixture skew -normal -symmetric Arellano-Valle et al (2006)

SE Skew-elliptical Azzalini amp Capitanio (1999 2003)

Arnold amp Beaver (2000a 2002)

Branco amp Dey (2001 2002)

Sahu Dey amp Branco (2003)

SL Skew-logistic Wahed amp Ali (2001)

SN Skew-normal Azzalini (1985 1986)

Azzalini amp Dalla Valle (1996)


Capitanio et al (2003)

Gupta amp Chen (2004)

SS Skew-symmetric Wang Boyer amp Genton (2004a)

SSL Skew-slash Wang amp Genton (2006)

ST Skew-t Azzalini amp Capitanio (2003)


SUE SUN Unified skew -elliptical -normal Arellano-Valle amp Azzalini (2006)

4 CONSTRUCTION OF SELECTION DISTRIBUTIONS

As was discussed in Section 22 whenP(U isin C |V = x) = P(U isin C) for all x we havein (2) thatfX = fV and so the original model is unaffected by the selection mechanism thatwe are considering WhenC = C(0) this fact can occur in many situations whereU andV are uncorrelated random vectors even though they are not independent In order to assurean explicit association betweenU andV we propose in this section several constructions ofexplicit selection mechanisms from which new classes of multivariate symmetric and skeweddistributions can be obtained We start with a random vectorZ isin IRk and consider two functionsu IRk rarr IRq andv IRk rarr IRp Denote these byU = u(Z) and byV = v(Z) We theninvestigate various selection distributions according to Definition 1 based onU andV and ameasurable subsetC of IRq


41 Linear selection distributions

The most important case of selection distributions is undoubtedly the one based on linear func-tionsu andv ie u(Z) = AZ + a andv(Z) = BZ + b whereA isin IRqtimesk andB isin IRptimesk

are matricesa isin IRq andb isin IRp are vectors and we assumep le k andq le k Under thisapproach it is natural to assume that the distribution ofZ is closed under linear transformationsZ sim ECk(0 Ik h(k)) say implying thatU = AZ + a andV = BZ + b will a have jointdistribution as in (7) withξ

U= a ξ

V= b ΩU = AA⊤ ΩV = BB⊤ and∆ = AB⊤ It

is then straightforward to establish the connection between (9) and the selection pdf resulting byusing the present approach We now examine the particular casefZ( middot ) = φk( middot ) the multivari-ate standard normal pdf for the random vectorZ isin IRk We further focus on subsetsC(β) of IRq

defined in (5) with the associated property that forY sim Nr(ξYΩY) with cdf Φr(y ξ

YΩY)

Φr(C(β) ξY

ΩY) = P(Y gt β) = P(minusY lt minusβ) = Φr(minusβminusξY

ΩY) = Φr(ξYβΩY)

(26)Using (5) and (26) the pdf (13) simplifies to

fX(x) = φp(xb BB⊤)Φq(a + AB⊤(BB⊤)minus1(x minus b)β AA⊤ minus AB⊤(BB⊤)minus1BA⊤)

Φq(aβ AA⊤)

(27)Apart from some differences in parameterization the skew-normal pdf in (27) is analogous to theunified skew-normal (SUN) pdf introduced recently by Arellano-Valle amp Azzalini (2006) As isshown by these authors from theSUNpdf most other existingSNprobability density functionscan be obtained by appropriate modifications in the parameterization

Another interesting application arises when we consider linear functions of an exchangeablerandom vectorZ which can be used to obtain the density of linear combinations of the orderstatistics obtained fromZ see Arellano-Valle amp Genton (2006ab)

42 Nonlinear selection distributions

Here we discuss two types of special constructions of a selection distribution by letting (i)V =g(UT) or (ii) U = g(VT) whereg is an appropriate Borel function possibly nonlinear andT isin IRr is a random vector independent of (or uncorrelated with)U or V depending on thesituation (i) or (ii) respectively We obtain stochastic representations as a by-product

Let V = g(UT) whereU T are independent (or uncorrelated) random vectors Forexample the most important case arises wheng is a linear function ieV = AU+BT In such

situation the independence assumption implies that(V |U isin C)d= A(U |U isin C)+BT which

is a stochastic representation of convolution type This type of representation jointly with theindependence assumption has been used by some authors to define skewed distributions (seeeg Arnold amp Beaver 2002) but considering the particular subset of the formC(β) given in(5) only However this definition does not include many other families of skewed distributionssuch as the class of unified skew-elliptical distributions orSUE introduced by Arellano-Valle ampAzzalini (2006) The latter class has also a convolution type representation for the caseC(0)and does not require the independence assumption In fact as was shown by Arellano del Pinoamp San Martın (2002) (see also Arellano-Valle amp del Pino 2004) the independence assumptioncan be relaxed by considering that the random vector(UT) is in theC-class (renamedSI-classin Arellano-Valle amp del Pino 2004) SpecificallyC is the class of all symmetric random vectorsZ with P(Z = 0) = 0 and such that|Z| = (|Z1| |Zm|)⊤ andsign(Z) = (W1 Wm)⊤

are independent andsign(Z) sim Um whereWi = +1 if Zi gt 0 andWi = minus1 if Zi lt 0i = 1 m andUm is the uniform distribution onminus1 1m In that case(g(UT) |U gt 0)d= g(U[0]T) whereU[c]

d= (U |U gt c) As we show in the next section other families of

asymmetric distributions such as those considered by Fernandez amp Steel (1998) Mudholkar ampHutson (2000) and Arellano-Valle Gomez amp Quintana (2005c) can also be viewed as selectiondistributions by using this framework


Now let U = g(VT) with similar conditions as above Linear and nonlinear truncatedconditions onV can be introduced by this approach An important example is when we considerthe functionU = w(V) minus T with w IRp rarr IRq and the independence assumption betweenV

andT In that case (2) withC = C(0) reduces to

fX(x) = fV(x)FT(w(x))

P(U gt 0) (28)

If U has a symmetric distribution (about0) thenP(U gt 0) = FU(0) which holds eg ifw(minusx) = minusw(x) andV andT are symmetric random vectors In particular ifU is a randomvector in theC-class thenFU(0) = 2minusq and (28) yields

fX(x) = 2qfV(x)FT(w(x)) (29)

which is an extension of a similar result given by Azzalini amp Capitanio (1999 2003) Notethat (28) is not altered if the independence condition betweenV andT is replaced by the con-dition thatV andT are conditionally independent givenw(V) The result can be extendedhowever for the case whereV andT are not independent by replacing the marginal cdfFT bythe conditional cdfFT |V=x Thus flexible families of multivariate skewed probability den-sity functions as described in Section 37 can be obtained from (28) or (29) For example ifT sim Nq(0 Iq) is independent ofV with a symmetric probability density functionfp thenU |V = x sim Nq(w(x) Iq) and its marginal distribution is symmetric sincew( middot ) is an oddfunction In that case (28) reduces toKminus1fp(x)Φq(w(x)) whereK = E[Φq(w(V))] Finallywe note that this approach was simply the starting point for Azzalini (1985) He introduced theunivariate family of probability density functions2f(x)G(λx) x isin IR λ isin IR by consideringthe random variableU = λV minus T with the assumption thatV andT are symmetric (about0)and independent random variables withfV = f andFT = G This implies thatU is a symmetricrandom variable (about0) and soP(U ge 0) = P(T le λV ) =

intinfin

minusinfinG(λv)f(v) dv = 12 for

any value ofλOn the other hand as we mentioned in the Introduction symmetric selection models can also

be obtained depending on the conditioning constraints For example ifV = RV0 is a spherical

random vector ieRd= V andV0

d= VV and they are independent then the selection

random vector(V| V isin C)d= (R |R isin C)V0 Its distribution is also spherical and reduces to

the original spherical distribution ie the distribution ofV whenC = C(0) Thus new familiesof symmetric distributions can be obtained from the general class of selection distributions andof course they can be used to obtain many more families of new skewed distributions

43 Other classes of skewed distributions

As was shown by Arellano-Valle Gomez amp Quintana (2005c) another very general family ofunivariate skewed distributions is defined by the pdf of a random variableX given by

fX(x) =2

a(α) + b(α)

[f

(x

a(α)

)Ix ge 0 + f

(x

b(α)

)Ix lt 0

] x isin IR (30)

whereI middot denotes the indicator functionf(minusx) = f(x) for all x isin IR α is an asymmetryparameter (possibly vectorial) anda(α) and b(α) are known and positive asymmetric func-tions The skewed class of Fernandez amp Steel (1998) can be obtained from (30) by choosinga(γ) = γ and b(γ) = 1γ for γ gt 0 In turn theε-skew-normal pdf considered in Mud-holkar amp Hutson (2000) follows whena(ε) = 1 minus ε and b(ε) = 1 + ε with |ε| lt 1 Aparticular characteristic of these families is that they all have the same mode as their parent sym-metric probability density functions Moreover Arellano-Valle Gomez amp Quintana (2005c)showed that (30) corresponds to the pdf ofX = Wα|Z| whereZ has pdff and is independentof Wα which is a discrete random variable withP(Wα = a(α)) = a(α)(a(α) + b(α)) and


P(Wα = minusb(α)) = b(α)(a(α) + b(α)) Thus since|Z|d= (Z |Z gt 0) from a selection point

of view (30) can be obtained also from (2) sinceXd= (V |U isin C) with V = WαZ U = Z

andC = C(0) For an arbitrary selection setC more general and flexible families of proba-bility density functions can be obtained following this approach Moreover their correspondinglocation-scale extensions can be derived by considering the usual transformationY = ξ + τXwith ξ isin IR andτ gt 0 Multivariate extensions when the above ingredients are replaced byvectorial quantities and the connection with theC-class are discussed in Arellano-Valle Gomezamp Quintana (2005c) Another approach for obtaining multivariate extensions by consideringnonsingular affine transformations of independent skewed random variables is discussed by Fer-reira amp Steel (2004) Finally a thorough discussion of the connection between the selectionframework and the fact that any random variable can be determined by its sign and its absolutevalue can be found in Arellano-Valle amp del Pino (2004)

5 DISCUSSION

In this article we have proposed a unified framework for selection distributions and have de-scribed some of their theoretical properties and applications However much more researchneeds to be carried out to investigate unexplored aspects of these distributions in particular in-ference Indeed the currently available models and their properties are mainly based on theselection subsetC(β) defined by (5) and summarized in the left panel of Figure 1 Azza-lini (2005) presents an excellent overview of this particular case of selection mechanisms thatcomplements our present article nicely However multivariate distributions resulting from otherselection mechanisms remain mostly unexplored see the right panel of Figure 1 The presentarticle serves as a unification of the theory of many known multivariate skewed distributionsGenton (2005) discusses several possibilities for combining semiparametric and nonparametrictechniques with selection distributions Arellano-Valle amp Azzalini (2006) gave a unified treat-ment of most existing multivariate skew-normal distributions and discussed extensions to theskew-elliptical class They also considered the case of the singular multivariate skew-normaldistribution where the key covariance matrixΩ does not have full rank

Selection distributions provide a useful mechanism for obtaining a joint statistical modelwhen we have additional information about the occurrence of some related random events Wehave illustrated this point with various examples There is an unlimited potential for other appli-cations necessitating similar models describing selection procedures For instance Chen (2004)describes applications of multivariate skewed distributions as a link function in discrete selectionmodels and Bazan Branco amp Bolfarine (2006) propose a new skewed family of models for itemresponse theory We expect to see an important growth of applications of selection distributionsin the near future

APPENDIX

Moment generating function ofSLCT -Npq distributions The moment generating function ofX sim SLCT -Npq follows directly from the form of (13) We have

MX(s) = E[exps⊤X]

=

intexp(s⊤x)φp(x ξ

VΩV)

Φq(C ∆⊤Ωminus1V

(x minus ξV

) + ξU

ΩU minus ∆⊤Ωminus1V

∆)

Φq(C ξU

ΩU)dx

= exp

s⊤ξ

V+

1

2s⊤ΩVs

1

Φq(C ξU

ΩU)

times

intφp(x ξ

V+ ΩVsΩV)Φq(C ∆⊤Ωminus1

V(x minus ξ

V) + ξ


V∆) dx

= exp

s⊤ξ

V+

1

2s⊤ΩVs

1

Φq(C ξU

ΩU)E[P(U isin C | V = x)]


whereU | V = x sim Nq(∆⊤Ωminus1

V(xminusξ

V)+ξ

UΩUminus∆⊤Ωminus1

V∆) andV sim Np(ξV

+ΩVsΩV)and thusU sim Nq(∆

⊤s+ξU

ΩU) The result is then obtained from the relationE[P(U isin C | V

= x)] = P(U isin C) = Φq(C∆⊤s + ξU

ΩU) 2

Moment generating function of scale mixture ofSLCT -Npq distributionsThe moment gener-ating function ofX such that(X |W = w) sim SLCT -Npq(ξ wΩ C(ξ

U)) whereW sim G

follows directly from the form of (17) and (14) We have

MX(s) = exps⊤ξVE[MXN

(radicW s

)]

= exps⊤ξV

intinfin

0exp1

2ws⊤ΩVsΦq

(radicw ∆⊤s0 wΩU

)dG(w)

Φq(00ΩU)

= exps⊤ξV

int infin

0

exp

w

2s⊤ΩVs

dG(w)

Φq(∆⊤s0ΩU)

Φq(00ΩU)

= E[MV|W (s)]Φq(∆

⊤s0ΩU)

Φq(00ΩU)

2

ACKNOWLEDGEMENTSThe authors are grateful to the Editor the Associate Editor and two anonymous referees for insightfulcomments on the manuscript This project was partially supported by a grant from the Fondo Nacional deDesarrollo Cientifico y Tecnologico in Chile The work of Branco was partially supported by the BrazilianConselho Nacional de Desenvolvimento Cientıfico and Tecnologico (CNPq) The work of Genton waspartially supported by a grant from the United States National Science Foundation

REFERENCES

R B Arellano-Valle amp A Azzalini (2006) On the unification of families of skew-normal distributionsScandinavian Journal of Statistics 33 561ndash574

R B Arellano-Valle amp G E del Pino (2004) From symmetric to asymmetric distributions a unified ap-proach InSkew-Elliptical Distributions and Their Applications A Journey Beyond Normality(M GGenton ed) Chapman amp HallCRC Boca Raton Florida pp 113ndash130

R B Arellano-Valle G E del Pino amp E San Martın (2002) Definition and probabilistic properties ofskew-distributionsStatistics amp Probability Letters 58 111ndash121

R B Arellano-Valle amp M G Genton (2005) On fundamental skew distributionsJournal of MultivariateAnalysis 96 93ndash116

R B Arellano-Valle amp M G Genton (2006a) On the exact distribution of the maximum of dependentrandom variablesStatistics amp Probability Letters revised version submitted for publication

R B Arellano-Valle amp M G Genton (2006b) On the exact distribution of linear combinations of orderstatistics from dependent random variablesJournal of Multivariate Analysis revised version submittedfor publication

R B Arellano-Valle M G Genton H W Gomez amp P Iglesias (2006) Shape mixtures of skewed distrib-utions with application in regression analysis Submitted for publication

R B Arellano-Valle H W Gomez amp F A Quintana (2004b) A new class of skew-normal distributionsCommunications in Statistics Theory and Methods 33 1465ndash1480

R B Arellano-Valle H W Gomez amp F A Quintana (2005c) Statistical inference for a general class ofasymmetric distributionsJournal of Statistical Planning and Inference 128 427ndash443

B C Arnold amp R J Beaver (2000a) Hidden truncation modelsSankya Series A 62 22ndash35

B C Arnold amp R J Beaver (2000b) The skew-Cauchy distributionStatistics amp Probability Letters 49285ndash290

B C Arnold amp R J Beaver (2002) Skewed multivariate models related to hidden truncation andor selec-tive reportingTest 11 7ndash54


B C Arnold amp R J Beaver (2004) Elliptical models subject to hidden truncation or selective samplingIn Skew-Elliptical Distributions and Their Applications A Journey Beyond Normality(M G Gentoned) Chapman amp HallCRC Boca Raton Florida pp 101ndash112

B C Arnold R J Beaver R A Groeneveld amp W Q Meeker (1993) The nontruncated marginal of atruncated bivariate normal distributionPsychometrika 58 471ndash478

B C Arnold E Castillo amp J M Sarabia (1999)Conditional Specification of Statistical Models Springer-Verlag New York

B C Arnold E Castillo amp J M Sarabia (2001) Conditionally specified distributions an introduction (withdiscussion)Statistical Science 16 249ndash274

A Azzalini (1985) A class of distributions which includes the normal onesScandinavian Journal of Sta-tistics 12 171ndash178

A Azzalini (1986) Further results on a class of distributions which includes the normal onesStatistica46 199ndash208

A Azzalini (2005) The skew-normal distribution and related multivariate familiesScandinavian Journalof Statistics 32 159ndash188

A Azzalini amp A Capitanio (1999) Statistical applications of the multivariate skew-normal distributionJournal of the Royal Statistical Society Series B 61 579ndash602

A Azzalini amp A Capitanio (2003) Distributions generated by perturbation of symmetry with emphasis ona multivariate skewt distributionJournal of the Royal Statistical Society Series B 65 367ndash389

A Azzalini amp A Dalla Valle (1996) The multivariate skew-normal distributionBiometrika 83 715ndash726

M J Bayarri amp M DeGroot (1992) A BAD view of weighted distributions and selection models InBayesian Statistics 4 Proceedings of the Fourth Valencia International Meeting April 15ndash20 1991(J M Bernardo J O Berger A P Dawid amp A F M Smith eds) Oxford University Press pp 17ndash29

J L Bazan M D Branco amp H Bolfarine (2006) A skew item response modelBayesian Analysis inpress

Z W Birnbaum (1950) Effect of linear truncation on a multinormal populationAnnals of MathematicalStatistics 21 272ndash279

M D Branco amp D K Dey (2001) A general class of multivariate skew-elliptical distributionsJournal ofMultivariate Analysis 79 99ndash113

M D Branco amp D K Dey (2002) Regression model under skew elliptical error distributionThe Journalof Mathematical Sciences 1 151-169

A Capitanio A Azzalini amp E Stanghellini (2003) Graphical models for skew-normal variatesScandina-vian Journal of Statistics 30 129ndash144

M-H Chen (2004) Skewed link models for categorical response data InSkew-Elliptical Distributions andTheir Applications A Journey Beyond Normality(M G Genton ed) Chapman amp HallCRC BocaRaton Florida pp 131ndash151

J B Copas amp H G Li (1997) Inference for non-random samples (with discussion)Journal of the RoyalStatistical Society Series B 59 55ndash95

C Crocetta amp N Loperfido (2005) The exact sampling distribution ofL-statisticsMetron 63 1ndash11

T J DiCiccio amp A C Monti (2004) Inferential aspects of the skew exponential power distributionJournalof the American Statistical Association 99 439ndash450

C Fernandez amp M F Steel (1998) On Bayesian modelling of fat tails and skewnessJournal of the Amer-ican Statistical Association 93 359ndash371

J T A S Ferreira amp M F J Steel (2004) Bayesian multivariate skewed regression modelling with anapplication to firm size InSkew-Elliptical Distributions and Their Applications A Journey BeyondNormality(M G Genton ed) Chapman amp HallCRC Boca Raton Florida pp 175ndash190

M G Genton ed (2004a)Skew-Elliptical Distributions and Their Applications A Journey Beyond Nor-mality Chapman amp HallCRC Boca Raton Florida

M G Genton (2004b) Skew-symmetric and generalized skew-elliptical distributions InSkew-EllipticalDistributions and Their Applications A Journey Beyond Normality(M G Genton ed) Chapman ampHallCRC Boca Raton Florida pp 81ndash100


M G Genton (2005) Discussion of The skew-normal distribution and related multivariate families by AAzzalini Scandinavian Journal of Statistics 32 189ndash198

M G Genton amp N Loperfido (2005) Generalized skew-elliptical distributions and their quadratic formsAnnals of the Institute of Statistical Mathematics 57 389ndash401

G Gonzalez-Farıas J A Domınguez-Molina amp A K Gupta (2004a) The closed skew-normal distributionIn Skew-Elliptical Distributions and Their Applications A Journey Beyond Normality(M G Gentoned) Chapman amp HallCRC Boca Raton Florida pp 25ndash42

G Gonzalez-Farıas J A Domınguez-Molina amp A K Gupta (2004b) Additive properties of skew normalrandom vectorsJournal of Statistical Planning and Inference 126 521ndash534

A K Gupta amp J T Chen (2004) A class of multivariate skew-normal modelsAnnals of the InstituteStatistical Mathematics 56 305ndash315

J Heckman (1979) Sample selection bias as a specification errorEconometrica 47 153ndash161

N L Johnson S Kotz amp N Balakrishnan (1994)Continuous Univariate Distributions 1 Wiley NewYork

B Liseo amp N Loperfido (2006) A note on reference priors for the scalar skew-normal distributionJournalof Statistical Planning and Inference 136 373ndash389

N Loperfido (2002) Statistical implications of selectively reported inferential resultsStatistics amp Proba-bility Letters 56 13ndash22

Y Ma amp M G Genton (2004) A flexible class of skew-symmetric distributionsScandinavian Journal ofStatistics 31 459ndash468

Y Ma M G Genton amp A A Tsiatis (2005) Locally efficient semiparametric estimators for generalizedskew-elliptical distributionsJournal of the American Statistical Association 100 980ndash989

Y Ma amp J Hart (2006) Constrained local likelihood estimators for semiparametric skew-normal distribu-tionsBiometrika in press

K G Malmquist (1920) A study of the stars of spectral type AMeddelande fran Lunds AstronomiskaObservatorium Serie II 22 1ndash39

G S Mudholkar amp A D Hutson (2000) The epsilon-skew-normal distribution for analyzing near-normaldataJournal of Statistical Planning and Inference 83 291ndash309

L S Nelson (1964) The sum of values from a normal and a truncated normal distributionTechnometrics6 469ndash471

A OrsquoHagan amp T Leonard (1976) Bayes estimation subject to uncertainty about parameter constraintsBiometrika 63 201ndash202

A Pewsey (2000) Problems of inference for Azzalinirsquos skew-normal distributionJournal of Applied Sta-tistics 27 859ndash870

A Pewsey (2006) Some observations on a simple means of generating skew distributions InAdvances inDistribution Theory Order Statistics and Inference (B C arnold N Balakrishnan E Castillo amp J-MSarabia eds) Birkhauser Boston pp 75ndash84

C R Rao (1985) Weighted distributions arising out of methods of ascertainment What populations doesa sample represent InA Celebration of Statistics The ISI Centenary Volume(A C Atkinson S EFienberg eds) Springer-Verlag New York pp 543ndash569

C Roberts (1966) A correlation model useful in the study of twinsJournal of the American StatisticalAssociation 61 1184ndash1190

S K Sahu D K Dey amp M Branco (2003) A new class of multivariate skew distributions with applicationto Bayesian regression modelsThe Canadian Journal of Statistics 31 129ndash150

N Sartori (2006) Bias prevention of maximum likelihood estimates for scalar skew normal and skewt

distributionsJournal of Statistical Planning and Inference 136 4259ndash4275

A Wahed amp M M Ali (2001) The skew-logistic distributionJournal of Statistical Research 35 71ndash80

J Wang J Boyer amp M G Genton (2004a) A skew-symmetric representation of multivariate distributionsStatistica Sinica 14 1259ndash1270

J Wang J Boyer amp M G Genton (2004b) A note on an equivalence between chi-square and generalizedskew-normal distributionsStatistics amp Probability Letters 66 395ndash398


J Wang amp M G Genton (2006) The multivariate skew-slash distributionJournal of Statistical Planningand Inference 136 209ndash220

M A Weinstein (1964) The sum of values from a normal and a truncated normal distributionTechnomet-rics 6 104ndash105

Received 21 December 2005 Reinaldo B ARELLANO-VALLE reivallematpuccl

Accepted 25 July 2006 Departamento de Estadıstica

Pontificia Universidad Catolica de Chile

Santiago 22 Chile

Marcia D BRANCOmbrancoimeuspbr

Departamento de Estatıstica

Universidade Sao Paulo

Sao Paulo SP 05315-970 Brasil

Marc G GENTONgentonstattamuedu

Department of Statistics

Texas AampM University

College Station TX 77843-3143 USA


parametric classes of nonnormal distributions particularly in models that can account for skew-ness and kurtosis A popular approach to model departure from normality consists of modifyinga symmetric (normal or heavy tailed) probability density function of a random variablevectorin a multiplicative fashion thereby introducing skewness This idea has been in the literaturefor a long time (eg Birnbaum 1950 Nelson 1964 Weinstein 1964 Roberts 1966 OrsquoHagan ampLeonard 1976) But it was Azzalini (1985 1986) who thoroughly implemented this idea forthe univariate normal distribution yielding the so-called skew-normal distribution An exten-sion to the multivariate case was then introduced by Azzalini amp Dalla Valle (1996) Statisticalapplications of the multivariate skew-normal distribution were presented by Azzalini amp Capi-tanio (1999) who also briefly discussed an extension to elliptical densities Arnold Castillo ampSarabia (1999 2001) discussed conditionally specified distributions related to the above con-struction for skewness

Since then many authors have tried to generalize these developments to skewing arbitrarysymmetric probability density functions with very general forms of multiplicative functionsSome of those proposals even allow for the construction of multimodal distributions In thepresent article we will provide an overview of existing distributions and novel extensions from aunified point of view

Besides the appeal of extending the classical multivariate normal theory the current impetusin this field of research is its potential for new applications For example a skewed version oftheGARCHmodel (generalized autoregressive conditional heteroscedasticity) in economics therepresentation of the shape of a human brain with skewed distributions in image analysis or askewed Kalman filter for the analysis of climatic time series are just but a few examples of thesemany applications see Genton (2004a) for those and more

12 Selection mechanisms

In addition to the attractive features of the above distributions for parametric modelling of mul-tivariate nonnormal distributions (a flexible parametric alternative to multivariate nonparametricdensity estimation) there is also a natural and important context where these distributions appearnamely selection models (see Heckman 1979 and more recently Copas amp Li 1997) Indeedconsider the model where a random vectorX isin IRp is distributed with symmetric probabilitydensity functionf(xθ) whereθ is a vector of unknown parameters The usual statistical analy-sis assumes that a random sampleX1 Xn from f(xθ) can be observed in order to makeinference aboutθ Nevertheless there are many situations where such a random sample mightnot be available for instance if it is too difficult or too costly to obtain If the probability den-sity function is distorted by some multiplicative nonnegative weight functionw(xθη) whereη denotes some vector of additional unknown parameters then the observed data is a randomsample from a weighted distribution (see eg Rao 1985) with probability density function

f(xθ)w(xθη)

Ew(Xθη) (1)

In particular if the observed data are obtained only from a selected portion of the populationof interest then (1) is called a selection model For example this can happen if the vectorX ofcharacteristics of a certain population is measured only for individuals who manifest a certaindisease due to cost or ethical reasons see the survey article by Bayarri amp DeGroot (1992) andreferences therein If the weight function satisfiesEw(Xθη) = 12 the resulting proba-bility density function (1) has a normalizing constant that does not depend on the parametersθ

andη A weight function with such a property can naturally occur when the selection criterionis such that a certain variable of interest is larger than its expected value given other variables

A simple example of selection is given by dependent maxima Consider two random vari-ables with bivariate normal distribution mean zero variance one and correlationρ The distri-bution of the maximum of those two random variables is exactly skew-normal (Roberts 1966

















21 Main properties




P(U isin C) (2)






fX(x) =

intC

fUV(ux) duintC

fU(u) du (3)









fX2 |X1=x1(x2) = fV2 |V1=x1

(x2)P(U isin C |V1 = x1V2 = x2)

P(U isin C |V1 = x1) (4)


X1d= (V1 |U isin C)


d= (V2 |U isin C)











U

V

)sim ECq+p

(ξ =

(ξU

ξV

)Ω =

(ΩU ∆⊤

∆ ΩV

) h(q+p)

) (7)

whereξU

isin IRq andξV









V(x minus ξ


V∆ h

(q)υ(x))


υ(x) = (xminusξV

)⊤Ωminus1V

(xminusξV



fr(y ξY


)⊤Ωminus1Y

(y minus ξY

))



ΩY h(r)) ie

F r(C ξY

ΩY h(r)) =

int

C

fr(y ξY

ΩY h(r)) dy


(V |U isin C)

fX(x) = fp(x ξV


V(x minus ξ

V) + ξ


V∆ h

(q)υ(x))

F q(C ξU

ΩU h(q)) (9)


fX(x) =

intC


fq(u ξU

ΩU h(q)) du (10)



fX(x) = tp(x ξV

ΩV ν)


(x minus ξV

) + ξU

ν + υ(x)


V∆] ν + p)

T q(C ξU

ΩU ν)


(radicu)





(U

V

)sim Nq+p

(ξ =

(ξU

ξV

)Ω =

(ΩU ∆⊤

∆ ΩV

)) (12)






fX(x) = φp(x ξV


V(x minus ξ

V) + ξ


V∆)

Φq(C ξU

ΩU) (13)


MX(s) = exp

s⊤ξ

V+

1

2s⊤ΩVs

Φq(C∆⊤s + ξ

UΩU)

Φq(C ξU

ΩU) (14)





(ξU



U]

d=



Xd= ξ

V+ ∆Ωminus1

UX0[C minus ξ

U] + X1 (15)


E(X) = ξV

+ ∆Ωminus1U

E(X0[C minus ξU

])



])Ωminus1U

∆⊤








(U

V


(ξ =

(ξU

ξV

) wΩ =

(wΩU w∆⊤

w∆ wΩV

)) (16)


intinfin





intinfin



fX(x) =

int infin

0

φp(x ξV


V(x minus ξ

V) + ξ


V∆)

Φq(C ξU

wΩU)dG(w)


fX(x) =

intC

intinfin


C

intinfin

0φq(u ξ

U wΩU) dG(w) du



Xd= ξ

V+

radicW XN (17)



V+ E





that


⊤s0ΩU)

Φq(00ΩU) (18)


V+ 1

2ws⊤ΩVs










P(U gt 0) (19)


U= 0

ξV







fX(x) = fp(x ξV

ΩV h(p))Fq(ξU

+ ∆⊤Ωminus1V

(x minus ξV


∆ h(q)υ(x))

Fq(ξU0ΩU h(q))

(21)



)⊤Ωminus1V

(xminusξV

)













V+ AX0[minusξ


U]


U) with




(micro

V

)sim Nq+p

((a

0

)

(Γ ΓA⊤

AΓ Θ + AΓA⊤




Φq(c0Γ) (22)











fX(x) = 2φ(x)Φ

(λ1xradic

1 + λ2x2

)



2)



fX(x) = 2fV(x)π(x) (23)






fX(x) = 2fV(x)H(PK(x)) (24)







fX(x) = 2fp(x ξV


V) h

(1)υ(x))





fX(x) = 2tp(x ξV

ΩV ν)T1

((ν + 1

ν + υ(x)

)12

α⊤(x minus ξV

) ν + 1

)



fX(x) = 2φp(x ξV


)) (25)





SLCT

COther subsetsβ

FUSE

FSS

SUN

CFUSNCSN

SMSN

SESUEFUSN

SE

GSE

SN

GSNFSN FSE

SS

HSN

q=1

FUSS C( )

etc

C( minus )

C( ) C(0)

α

α

β

α

SMSS














FUSS -symmetric

























U= a ξ




YΩY)

Φr(C(β) ξY





Φq(aβ AA⊤)















P(U gt 0) (28)


fX(x) = 2qfV(x)FT(w(x)) (29)


intinfin










fX(x) =2

a(α) + b(α)

[f

(x

a(α)

)Ix ge 0 + f

(x

b(α)

)Ix lt 0

] x isin IR (30)






5 DISCUSSION



APPENDIX


MX(s) = E[exps⊤X]

=


VΩV)


(x minus ξV

) + ξU


∆)

Φq(C ξU

ΩU)dx

= exp

s⊤ξ

V+

1

2s⊤ΩVs

1

Φq(C ξU

ΩU)

times

intφp(x ξ


V(x minus ξ

V) + ξ


V∆) dx

= exp

s⊤ξ

V+

1

2s⊤ΩVs

1

Φq(C ξU




V(xminusξ

V)+ξ




⊤s+ξU



ΩU) 2


U)) whereW sim G



(radicW s

)]

= exps⊤ξV

intinfin

0exp1

2ws⊤ΩVsΦq


)dG(w)

Φq(00ΩU)

= exps⊤ξV

int infin

0

exp

w

2s⊤ΩVs

dG(w)

Φq(∆⊤s0ΩU)

Φq(00ΩU)


⊤s0ΩU)

Φq(00ΩU)

2


REFERENCES







































































Santiago 22 Chile

























21 Main properties




P(U isin C) (2)






fX(x) =

intC

fUV(ux) duintC

fU(u) du (3)









fX2 |X1=x1(x2) = fV2 |V1=x1

(x2)P(U isin C |V1 = x1V2 = x2)

P(U isin C |V1 = x1) (4)


X1d= (V1 |U isin C)


d= (V2 |U isin C)











U

V

)sim ECq+p

(ξ =

(ξU

ξV

)Ω =

(ΩU ∆⊤

∆ ΩV

) h(q+p)

) (7)

whereξU

isin IRq andξV









V(x minus ξ


V∆ h

(q)υ(x))


υ(x) = (xminusξV

)⊤Ωminus1V

(xminusξV



fr(y ξY


)⊤Ωminus1Y

(y minus ξY

))



ΩY h(r)) ie

F r(C ξY

ΩY h(r)) =

int

C

fr(y ξY

ΩY h(r)) dy


(V |U isin C)

fX(x) = fp(x ξV


V(x minus ξ

V) + ξ


V∆ h

(q)υ(x))

F q(C ξU

ΩU h(q)) (9)


fX(x) =

intC


fq(u ξU

ΩU h(q)) du (10)



fX(x) = tp(x ξV

ΩV ν)


(x minus ξV

) + ξU

ν + υ(x)


V∆] ν + p)

T q(C ξU

ΩU ν)


(radicu)





(U

V

)sim Nq+p

(ξ =

(ξU

ξV

)Ω =

(ΩU ∆⊤

∆ ΩV

)) (12)






fX(x) = φp(x ξV


V(x minus ξ

V) + ξ


V∆)

Φq(C ξU

ΩU) (13)


MX(s) = exp

s⊤ξ

V+

1

2s⊤ΩVs

Φq(C∆⊤s + ξ

UΩU)

Φq(C ξU

ΩU) (14)





(ξU



U]

d=



Xd= ξ

V+ ∆Ωminus1

UX0[C minus ξ

U] + X1 (15)


E(X) = ξV

+ ∆Ωminus1U

E(X0[C minus ξU

])



])Ωminus1U

∆⊤








(U

V


(ξ =

(ξU

ξV

) wΩ =

(wΩU w∆⊤

w∆ wΩV

)) (16)


intinfin





intinfin



fX(x) =

int infin

0

φp(x ξV


V(x minus ξ

V) + ξ


V∆)

Φq(C ξU

wΩU)dG(w)


fX(x) =

intC

intinfin


C

intinfin

0φq(u ξ

U wΩU) dG(w) du



Xd= ξ

V+

radicW XN (17)



V+ E





that


⊤s0ΩU)

Φq(00ΩU) (18)


V+ 1

2ws⊤ΩVs










P(U gt 0) (19)


U= 0

ξV







fX(x) = fp(x ξV

ΩV h(p))Fq(ξU

+ ∆⊤Ωminus1V

(x minus ξV


∆ h(q)υ(x))

Fq(ξU0ΩU h(q))

(21)



)⊤Ωminus1V

(xminusξV

)













V+ AX0[minusξ


U]


U) with




(micro

V

)sim Nq+p

((a

0

)

(Γ ΓA⊤

AΓ Θ + AΓA⊤




Φq(c0Γ) (22)











fX(x) = 2φ(x)Φ

(λ1xradic

1 + λ2x2

)



2)



fX(x) = 2fV(x)π(x) (23)






fX(x) = 2fV(x)H(PK(x)) (24)







fX(x) = 2fp(x ξV


V) h

(1)υ(x))





fX(x) = 2tp(x ξV

ΩV ν)T1

((ν + 1

ν + υ(x)

)12

α⊤(x minus ξV

) ν + 1

)



fX(x) = 2φp(x ξV


)) (25)





SLCT

COther subsetsβ

FUSE

FSS

SUN

CFUSNCSN

SMSN

SESUEFUSN

SE

GSE

SN

GSNFSN FSE

SS

HSN

q=1

FUSS C( )

etc

C( minus )

C( ) C(0)

α

α

β

α

SMSS














FUSS -symmetric

























U= a ξ




YΩY)

Φr(C(β) ξY





Φq(aβ AA⊤)















P(U gt 0) (28)


fX(x) = 2qfV(x)FT(w(x)) (29)


intinfin










fX(x) =2

a(α) + b(α)

[f

(x

a(α)

)Ix ge 0 + f

(x

b(α)

)Ix lt 0

] x isin IR (30)






5 DISCUSSION



APPENDIX


MX(s) = E[exps⊤X]

=


VΩV)


(x minus ξV

) + ξU


∆)

Φq(C ξU

ΩU)dx

= exp

s⊤ξ

V+

1

2s⊤ΩVs

1

Φq(C ξU

ΩU)

times

intφp(x ξ


V(x minus ξ

V) + ξ


V∆) dx

= exp

s⊤ξ

V+

1

2s⊤ΩVs

1

Φq(C ξU




V(xminusξ

V)+ξ




⊤s+ξU



ΩU) 2


U)) whereW sim G



(radicW s

)]

= exps⊤ξV

intinfin

0exp1

2ws⊤ΩVsΦq


)dG(w)

Φq(00ΩU)

= exps⊤ξV

int infin

0

exp

w

2s⊤ΩVs

dG(w)

Φq(∆⊤s0ΩU)

Φq(00ΩU)


⊤s0ΩU)

Φq(00ΩU)

2


REFERENCES







































































Santiago 22 Chile
















fX2 |X1=x1(x2) = fV2 |V1=x1

(x2)P(U isin C |V1 = x1V2 = x2)

P(U isin C |V1 = x1) (4)


X1d= (V1 |U isin C)


d= (V2 |U isin C)











U

V

)sim ECq+p

(ξ =

(ξU

ξV

)Ω =

(ΩU ∆⊤

∆ ΩV

) h(q+p)

) (7)

whereξU

isin IRq andξV









V(x minus ξ


V∆ h

(q)υ(x))


υ(x) = (xminusξV

)⊤Ωminus1V

(xminusξV



fr(y ξY


)⊤Ωminus1Y

(y minus ξY

))



ΩY h(r)) ie

F r(C ξY

ΩY h(r)) =

int

C

fr(y ξY

ΩY h(r)) dy


(V |U isin C)

fX(x) = fp(x ξV


V(x minus ξ

V) + ξ


V∆ h

(q)υ(x))

F q(C ξU

ΩU h(q)) (9)


fX(x) =

intC


fq(u ξU

ΩU h(q)) du (10)



fX(x) = tp(x ξV

ΩV ν)


(x minus ξV

) + ξU

ν + υ(x)


V∆] ν + p)

T q(C ξU

ΩU ν)


(radicu)





(U

V

)sim Nq+p

(ξ =

(ξU

ξV

)Ω =

(ΩU ∆⊤

∆ ΩV

)) (12)






fX(x) = φp(x ξV


V(x minus ξ

V) + ξ


V∆)

Φq(C ξU

ΩU) (13)


MX(s) = exp

s⊤ξ

V+

1

2s⊤ΩVs

Φq(C∆⊤s + ξ

UΩU)

Φq(C ξU

ΩU) (14)





(ξU



U]

d=



Xd= ξ

V+ ∆Ωminus1

UX0[C minus ξ

U] + X1 (15)


E(X) = ξV

+ ∆Ωminus1U

E(X0[C minus ξU

])



])Ωminus1U

∆⊤








(U

V


(ξ =

(ξU

ξV

) wΩ =

(wΩU w∆⊤

w∆ wΩV

)) (16)


intinfin





intinfin



fX(x) =

int infin

0

φp(x ξV


V(x minus ξ

V) + ξ


V∆)

Φq(C ξU

wΩU)dG(w)


fX(x) =

intC

intinfin


C

intinfin

0φq(u ξ

U wΩU) dG(w) du



Xd= ξ

V+

radicW XN (17)



V+ E





that


⊤s0ΩU)

Φq(00ΩU) (18)


V+ 1

2ws⊤ΩVs










P(U gt 0) (19)


U= 0

ξV







fX(x) = fp(x ξV

ΩV h(p))Fq(ξU

+ ∆⊤Ωminus1V

(x minus ξV


∆ h(q)υ(x))

Fq(ξU0ΩU h(q))

(21)



)⊤Ωminus1V

(xminusξV

)













V+ AX0[minusξ


U]


U) with




(micro

V

)sim Nq+p

((a

0

)

(Γ ΓA⊤

AΓ Θ + AΓA⊤




Φq(c0Γ) (22)











fX(x) = 2φ(x)Φ

(λ1xradic

1 + λ2x2

)



2)



fX(x) = 2fV(x)π(x) (23)






fX(x) = 2fV(x)H(PK(x)) (24)







fX(x) = 2fp(x ξV


V) h

(1)υ(x))





fX(x) = 2tp(x ξV

ΩV ν)T1

((ν + 1

ν + υ(x)

)12

α⊤(x minus ξV

) ν + 1

)



fX(x) = 2φp(x ξV


)) (25)





SLCT

COther subsetsβ

FUSE

FSS

SUN

CFUSNCSN

SMSN

SESUEFUSN

SE

GSE

SN

GSNFSN FSE

SS

HSN

q=1

FUSS C( )

etc

C( minus )

C( ) C(0)

α

α

β

α

SMSS














FUSS -symmetric

























U= a ξ




YΩY)

Φr(C(β) ξY





Φq(aβ AA⊤)















P(U gt 0) (28)


fX(x) = 2qfV(x)FT(w(x)) (29)


intinfin










fX(x) =2

a(α) + b(α)

[f

(x

a(α)

)Ix ge 0 + f

(x

b(α)

)Ix lt 0

] x isin IR (30)






5 DISCUSSION



APPENDIX


MX(s) = E[exps⊤X]

=


VΩV)


(x minus ξV

) + ξU


∆)

Φq(C ξU

ΩU)dx

= exp

s⊤ξ

V+

1

2s⊤ΩVs

1

Φq(C ξU

ΩU)

times

intφp(x ξ


V(x minus ξ

V) + ξ


V∆) dx

= exp

s⊤ξ

V+

1

2s⊤ΩVs

1

Φq(C ξU




V(xminusξ

V)+ξ




⊤s+ξU



ΩU) 2


U)) whereW sim G



(radicW s

)]

= exps⊤ξV

intinfin

0exp1

2ws⊤ΩVsΦq


)dG(w)

Φq(00ΩU)

= exps⊤ξV

int infin

0

exp

w

2s⊤ΩVs

dG(w)

Φq(∆⊤s0ΩU)

Φq(00ΩU)


⊤s0ΩU)

Φq(00ΩU)

2


REFERENCES







































































Santiago 22 Chile















V(x minus ξ


V∆ h

(q)υ(x))


υ(x) = (xminusξV

)⊤Ωminus1V

(xminusξV



fr(y ξY


)⊤Ωminus1Y

(y minus ξY

))



ΩY h(r)) ie

F r(C ξY

ΩY h(r)) =

int

C

fr(y ξY

ΩY h(r)) dy


(V |U isin C)

fX(x) = fp(x ξV


V(x minus ξ

V) + ξ


V∆ h

(q)υ(x))

F q(C ξU

ΩU h(q)) (9)


fX(x) =

intC


fq(u ξU

ΩU h(q)) du (10)



fX(x) = tp(x ξV

ΩV ν)


(x minus ξV

) + ξU

ν + υ(x)


V∆] ν + p)

T q(C ξU

ΩU ν)


(radicu)





(U

V

)sim Nq+p

(ξ =

(ξU

ξV

)Ω =

(ΩU ∆⊤

∆ ΩV

)) (12)






fX(x) = φp(x ξV


V(x minus ξ

V) + ξ


V∆)

Φq(C ξU

ΩU) (13)


MX(s) = exp

s⊤ξ

V+

1

2s⊤ΩVs

Φq(C∆⊤s + ξ

UΩU)

Φq(C ξU

ΩU) (14)





(ξU



U]

d=



Xd= ξ

V+ ∆Ωminus1

UX0[C minus ξ

U] + X1 (15)


E(X) = ξV

+ ∆Ωminus1U

E(X0[C minus ξU

])



])Ωminus1U

∆⊤








(U

V


(ξ =

(ξU

ξV

) wΩ =

(wΩU w∆⊤

w∆ wΩV

)) (16)


intinfin





intinfin



fX(x) =

int infin

0

φp(x ξV


V(x minus ξ

V) + ξ


V∆)

Φq(C ξU

wΩU)dG(w)


fX(x) =

intC

intinfin


C

intinfin

0φq(u ξ

U wΩU) dG(w) du



Xd= ξ

V+

radicW XN (17)



V+ E





that


⊤s0ΩU)

Φq(00ΩU) (18)


V+ 1

2ws⊤ΩVs










P(U gt 0) (19)


U= 0

ξV







fX(x) = fp(x ξV

ΩV h(p))Fq(ξU

+ ∆⊤Ωminus1V

(x minus ξV


∆ h(q)υ(x))

Fq(ξU0ΩU h(q))

(21)



)⊤Ωminus1V

(xminusξV

)













V+ AX0[minusξ


U]


U) with




(micro

V

)sim Nq+p

((a

0

)

(Γ ΓA⊤

AΓ Θ + AΓA⊤




Φq(c0Γ) (22)











fX(x) = 2φ(x)Φ

(λ1xradic

1 + λ2x2

)



2)



fX(x) = 2fV(x)π(x) (23)






fX(x) = 2fV(x)H(PK(x)) (24)







fX(x) = 2fp(x ξV


V) h

(1)υ(x))





fX(x) = 2tp(x ξV

ΩV ν)T1

((ν + 1

ν + υ(x)

)12

α⊤(x minus ξV

) ν + 1

)



fX(x) = 2φp(x ξV


)) (25)





SLCT

COther subsetsβ

FUSE

FSS

SUN

CFUSNCSN

SMSN

SESUEFUSN

SE

GSE

SN

GSNFSN FSE

SS

HSN

q=1

FUSS C( )

etc

C( minus )

C( ) C(0)

α

α

β

α

SMSS














FUSS -symmetric

























U= a ξ




YΩY)

Φr(C(β) ξY





Φq(aβ AA⊤)















P(U gt 0) (28)


fX(x) = 2qfV(x)FT(w(x)) (29)


intinfin










fX(x) =2

a(α) + b(α)

[f

(x

a(α)

)Ix ge 0 + f

(x

b(α)

)Ix lt 0

] x isin IR (30)






5 DISCUSSION



APPENDIX


MX(s) = E[exps⊤X]

=


VΩV)


(x minus ξV

) + ξU


∆)

Φq(C ξU

ΩU)dx

= exp

s⊤ξ

V+

1

2s⊤ΩVs

1

Φq(C ξU

ΩU)

times

intφp(x ξ


V(x minus ξ

V) + ξ


V∆) dx

= exp

s⊤ξ

V+

1

2s⊤ΩVs

1

Φq(C ξU




V(xminusξ

V)+ξ




⊤s+ξU



ΩU) 2


U)) whereW sim G



(radicW s

)]

= exps⊤ξV

intinfin

0exp1

2ws⊤ΩVsΦq


)dG(w)

Φq(00ΩU)

= exps⊤ξV

int infin

0

exp

w

2s⊤ΩVs

dG(w)

Φq(∆⊤s0ΩU)

Φq(00ΩU)


⊤s0ΩU)

Φq(00ΩU)

2


REFERENCES







































































Santiago 22 Chile












(U

V

)sim Nq+p

(ξ =

(ξU

ξV

)Ω =

(ΩU ∆⊤

∆ ΩV

)) (12)






fX(x) = φp(x ξV


V(x minus ξ

V) + ξ


V∆)

Φq(C ξU

ΩU) (13)


MX(s) = exp

s⊤ξ

V+

1

2s⊤ΩVs

Φq(C∆⊤s + ξ

UΩU)

Φq(C ξU

ΩU) (14)





(ξU



U]

d=



Xd= ξ

V+ ∆Ωminus1

UX0[C minus ξ

U] + X1 (15)


E(X) = ξV

+ ∆Ωminus1U

E(X0[C minus ξU

])



])Ωminus1U

∆⊤








(U

V


(ξ =

(ξU

ξV

) wΩ =

(wΩU w∆⊤

w∆ wΩV

)) (16)


intinfin





intinfin



fX(x) =

int infin

0

φp(x ξV


V(x minus ξ

V) + ξ


V∆)

Φq(C ξU

wΩU)dG(w)


fX(x) =

intC

intinfin


C

intinfin

0φq(u ξ

U wΩU) dG(w) du



Xd= ξ

V+

radicW XN (17)



V+ E





that


⊤s0ΩU)

Φq(00ΩU) (18)


V+ 1

2ws⊤ΩVs










P(U gt 0) (19)


U= 0

ξV







fX(x) = fp(x ξV

ΩV h(p))Fq(ξU

+ ∆⊤Ωminus1V

(x minus ξV


∆ h(q)υ(x))

Fq(ξU0ΩU h(q))

(21)



)⊤Ωminus1V

(xminusξV

)













V+ AX0[minusξ


U]


U) with




(micro

V

)sim Nq+p

((a

0

)

(Γ ΓA⊤

AΓ Θ + AΓA⊤




Φq(c0Γ) (22)











fX(x) = 2φ(x)Φ

(λ1xradic

1 + λ2x2

)



2)



fX(x) = 2fV(x)π(x) (23)






fX(x) = 2fV(x)H(PK(x)) (24)







fX(x) = 2fp(x ξV


V) h

(1)υ(x))





fX(x) = 2tp(x ξV

ΩV ν)T1

((ν + 1

ν + υ(x)

)12

α⊤(x minus ξV

) ν + 1

)



fX(x) = 2φp(x ξV


)) (25)





SLCT

COther subsetsβ

FUSE

FSS

SUN

CFUSNCSN

SMSN

SESUEFUSN

SE

GSE

SN

GSNFSN FSE

SS

HSN

q=1

FUSS C( )

etc

C( minus )

C( ) C(0)

α

α

β

α

SMSS














FUSS -symmetric

























U= a ξ




YΩY)

Φr(C(β) ξY





Φq(aβ AA⊤)















P(U gt 0) (28)


fX(x) = 2qfV(x)FT(w(x)) (29)


intinfin










fX(x) =2

a(α) + b(α)

[f

(x

a(α)

)Ix ge 0 + f

(x

b(α)

)Ix lt 0

] x isin IR (30)






5 DISCUSSION



APPENDIX


MX(s) = E[exps⊤X]

=


VΩV)


(x minus ξV

) + ξU


∆)

Φq(C ξU

ΩU)dx

= exp

s⊤ξ

V+

1

2s⊤ΩVs

1

Φq(C ξU

ΩU)

times

intφp(x ξ


V(x minus ξ

V) + ξ


V∆) dx

= exp

s⊤ξ

V+

1

2s⊤ΩVs

1

Φq(C ξU




V(xminusξ

V)+ξ




⊤s+ξU



ΩU) 2


U)) whereW sim G



(radicW s

)]

= exps⊤ξV

intinfin

0exp1

2ws⊤ΩVsΦq


)dG(w)

Φq(00ΩU)

= exps⊤ξV

int infin

0

exp

w

2s⊤ΩVs

dG(w)

Φq(∆⊤s0ΩU)

Φq(00ΩU)


⊤s0ΩU)

Φq(00ΩU)

2


REFERENCES







































































Santiago 22 Chile












(U

V


(ξ =

(ξU

ξV

) wΩ =

(wΩU w∆⊤

w∆ wΩV

)) (16)


intinfin





intinfin



fX(x) =

int infin

0

φp(x ξV


V(x minus ξ

V) + ξ


V∆)

Φq(C ξU

wΩU)dG(w)


fX(x) =

intC

intinfin


C

intinfin

0φq(u ξ

U wΩU) dG(w) du



Xd= ξ

V+

radicW XN (17)



V+ E





that


⊤s0ΩU)

Φq(00ΩU) (18)


V+ 1

2ws⊤ΩVs










P(U gt 0) (19)


U= 0

ξV







fX(x) = fp(x ξV

ΩV h(p))Fq(ξU

+ ∆⊤Ωminus1V

(x minus ξV


∆ h(q)υ(x))

Fq(ξU0ΩU h(q))

(21)



)⊤Ωminus1V

(xminusξV

)













V+ AX0[minusξ


U]


U) with




(micro

V

)sim Nq+p

((a

0

)

(Γ ΓA⊤

AΓ Θ + AΓA⊤




Φq(c0Γ) (22)











fX(x) = 2φ(x)Φ

(λ1xradic

1 + λ2x2

)



2)



fX(x) = 2fV(x)π(x) (23)






fX(x) = 2fV(x)H(PK(x)) (24)







fX(x) = 2fp(x ξV


V) h

(1)υ(x))





fX(x) = 2tp(x ξV

ΩV ν)T1

((ν + 1

ν + υ(x)

)12

α⊤(x minus ξV

) ν + 1

)



fX(x) = 2φp(x ξV


)) (25)





SLCT

COther subsetsβ

FUSE

FSS

SUN

CFUSNCSN

SMSN

SESUEFUSN

SE

GSE

SN

GSNFSN FSE

SS

HSN

q=1

FUSS C( )

etc

C( minus )

C( ) C(0)

α

α

β

α

SMSS














FUSS -symmetric

























U= a ξ




YΩY)

Φr(C(β) ξY





Φq(aβ AA⊤)















P(U gt 0) (28)


fX(x) = 2qfV(x)FT(w(x)) (29)


intinfin










fX(x) =2

a(α) + b(α)

[f

(x

a(α)

)Ix ge 0 + f

(x

b(α)

)Ix lt 0

] x isin IR (30)






5 DISCUSSION



APPENDIX


MX(s) = E[exps⊤X]

=


VΩV)


(x minus ξV

) + ξU


∆)

Φq(C ξU

ΩU)dx

= exp

s⊤ξ

V+

1

2s⊤ΩVs

1

Φq(C ξU

ΩU)

times

intφp(x ξ


V(x minus ξ

V) + ξ


V∆) dx

= exp

s⊤ξ

V+

1

2s⊤ΩVs

1

Φq(C ξU




V(xminusξ

V)+ξ




⊤s+ξU



ΩU) 2


U)) whereW sim G



(radicW s

)]

= exps⊤ξV

intinfin

0exp1

2ws⊤ΩVsΦq


)dG(w)

Φq(00ΩU)

= exps⊤ξV

int infin

0

exp

w

2s⊤ΩVs

dG(w)

Φq(∆⊤s0ΩU)

Φq(00ΩU)


⊤s0ΩU)

Φq(00ΩU)

2


REFERENCES







































































Santiago 22 Chile














P(U gt 0) (19)


U= 0

ξV







fX(x) = fp(x ξV

ΩV h(p))Fq(ξU

+ ∆⊤Ωminus1V

(x minus ξV


∆ h(q)υ(x))

Fq(ξU0ΩU h(q))

(21)



)⊤Ωminus1V

(xminusξV

)













V+ AX0[minusξ


U]


U) with




(micro

V

)sim Nq+p

((a

0

)

(Γ ΓA⊤

AΓ Θ + AΓA⊤




Φq(c0Γ) (22)











fX(x) = 2φ(x)Φ

(λ1xradic

1 + λ2x2

)



2)



fX(x) = 2fV(x)π(x) (23)






fX(x) = 2fV(x)H(PK(x)) (24)







fX(x) = 2fp(x ξV


V) h

(1)υ(x))





fX(x) = 2tp(x ξV

ΩV ν)T1

((ν + 1

ν + υ(x)

)12

α⊤(x minus ξV

) ν + 1

)



fX(x) = 2φp(x ξV


)) (25)





SLCT

COther subsetsβ

FUSE

FSS

SUN

CFUSNCSN

SMSN

SESUEFUSN

SE

GSE

SN

GSNFSN FSE

SS

HSN

q=1

FUSS C( )

etc

C( minus )

C( ) C(0)

α

α

β

α

SMSS














FUSS -symmetric

























U= a ξ




YΩY)

Φr(C(β) ξY





Φq(aβ AA⊤)















P(U gt 0) (28)


fX(x) = 2qfV(x)FT(w(x)) (29)


intinfin










fX(x) =2

a(α) + b(α)

[f

(x

a(α)

)Ix ge 0 + f

(x

b(α)

)Ix lt 0

] x isin IR (30)






5 DISCUSSION



APPENDIX


MX(s) = E[exps⊤X]

=


VΩV)


(x minus ξV

) + ξU


∆)

Φq(C ξU

ΩU)dx

= exp

s⊤ξ

V+

1

2s⊤ΩVs

1

Φq(C ξU

ΩU)

times

intφp(x ξ


V(x minus ξ

V) + ξ


V∆) dx

= exp

s⊤ξ

V+

1

2s⊤ΩVs

1

Φq(C ξU




V(xminusξ

V)+ξ




⊤s+ξU



ΩU) 2


U)) whereW sim G



(radicW s

)]

= exps⊤ξV

intinfin

0exp1

2ws⊤ΩVsΦq


)dG(w)

Φq(00ΩU)

= exps⊤ξV

int infin

0

exp

w

2s⊤ΩVs

dG(w)

Φq(∆⊤s0ΩU)

Φq(00ΩU)


⊤s0ΩU)

Φq(00ΩU)

2


REFERENCES







































































Santiago 22 Chile

















V+ AX0[minusξ


U]


U) with




(micro

V

)sim Nq+p

((a

0

)

(Γ ΓA⊤

AΓ Θ + AΓA⊤




Φq(c0Γ) (22)











fX(x) = 2φ(x)Φ

(λ1xradic

1 + λ2x2

)



2)



fX(x) = 2fV(x)π(x) (23)






fX(x) = 2fV(x)H(PK(x)) (24)







fX(x) = 2fp(x ξV


V) h

(1)υ(x))





fX(x) = 2tp(x ξV

ΩV ν)T1

((ν + 1

ν + υ(x)

)12

α⊤(x minus ξV

) ν + 1

)



fX(x) = 2φp(x ξV


)) (25)





SLCT

COther subsetsβ

FUSE

FSS

SUN

CFUSNCSN

SMSN

SESUEFUSN

SE

GSE

SN

GSNFSN FSE

SS

HSN

q=1

FUSS C( )

etc

C( minus )

C( ) C(0)

α

α

β

α

SMSS














FUSS -symmetric

























U= a ξ




YΩY)

Φr(C(β) ξY





Φq(aβ AA⊤)















P(U gt 0) (28)


fX(x) = 2qfV(x)FT(w(x)) (29)


intinfin










fX(x) =2

a(α) + b(α)

[f

(x

a(α)

)Ix ge 0 + f

(x

b(α)

)Ix lt 0

] x isin IR (30)






5 DISCUSSION



APPENDIX


MX(s) = E[exps⊤X]

=


VΩV)


(x minus ξV

) + ξU


∆)

Φq(C ξU

ΩU)dx

= exp

s⊤ξ

V+

1

2s⊤ΩVs

1

Φq(C ξU

ΩU)

times

intφp(x ξ


V(x minus ξ

V) + ξ


V∆) dx

= exp

s⊤ξ

V+

1

2s⊤ΩVs

1

Φq(C ξU




V(xminusξ

V)+ξ




⊤s+ξU



ΩU) 2


U)) whereW sim G



(radicW s

)]

= exps⊤ξV

intinfin

0exp1

2ws⊤ΩVsΦq


)dG(w)

Φq(00ΩU)

= exps⊤ξV

int infin

0

exp

w

2s⊤ΩVs

dG(w)

Φq(∆⊤s0ΩU)

Φq(00ΩU)


⊤s0ΩU)

Φq(00ΩU)

2


REFERENCES







































































Santiago 22 Chile













fX(x) = 2φ(x)Φ

(λ1xradic

1 + λ2x2

)



2)



fX(x) = 2fV(x)π(x) (23)






fX(x) = 2fV(x)H(PK(x)) (24)







fX(x) = 2fp(x ξV


V) h

(1)υ(x))





fX(x) = 2tp(x ξV

ΩV ν)T1

((ν + 1

ν + υ(x)

)12

α⊤(x minus ξV

) ν + 1

)



fX(x) = 2φp(x ξV


)) (25)





SLCT

COther subsetsβ

FUSE

FSS

SUN

CFUSNCSN

SMSN

SESUEFUSN

SE

GSE

SN

GSNFSN FSE

SS

HSN

q=1

FUSS C( )

etc

C( minus )

C( ) C(0)

α

α

β

α

SMSS














FUSS -symmetric

























U= a ξ




YΩY)

Φr(C(β) ξY





Φq(aβ AA⊤)















P(U gt 0) (28)


fX(x) = 2qfV(x)FT(w(x)) (29)


intinfin










fX(x) =2

a(α) + b(α)

[f

(x

a(α)

)Ix ge 0 + f

(x

b(α)

)Ix lt 0

] x isin IR (30)






5 DISCUSSION



APPENDIX


MX(s) = E[exps⊤X]

=


VΩV)


(x minus ξV

) + ξU


∆)

Φq(C ξU

ΩU)dx

= exp

s⊤ξ

V+

1

2s⊤ΩVs

1

Φq(C ξU

ΩU)

times

intφp(x ξ


V(x minus ξ

V) + ξ


V∆) dx

= exp

s⊤ξ

V+

1

2s⊤ΩVs

1

Φq(C ξU




V(xminusξ

V)+ξ




⊤s+ξU



ΩU) 2


U)) whereW sim G



(radicW s

)]

= exps⊤ξV

intinfin

0exp1

2ws⊤ΩVsΦq


)dG(w)

Φq(00ΩU)

= exps⊤ξV

int infin

0

exp

w

2s⊤ΩVs

dG(w)

Φq(∆⊤s0ΩU)

Φq(00ΩU)


⊤s0ΩU)

Φq(00ΩU)

2


REFERENCES







































































Santiago 22 Chile














fX(x) = 2fp(x ξV


V) h

(1)υ(x))





fX(x) = 2tp(x ξV

ΩV ν)T1

((ν + 1

ν + υ(x)

)12

α⊤(x minus ξV

) ν + 1

)



fX(x) = 2φp(x ξV


)) (25)





SLCT

COther subsetsβ

FUSE

FSS

SUN

CFUSNCSN

SMSN

SESUEFUSN

SE

GSE

SN

GSNFSN FSE

SS

HSN

q=1

FUSS C( )

etc

C( minus )

C( ) C(0)

α

α

β

α

SMSS














FUSS -symmetric

























U= a ξ




YΩY)

Φr(C(β) ξY





Φq(aβ AA⊤)















P(U gt 0) (28)


fX(x) = 2qfV(x)FT(w(x)) (29)


intinfin










fX(x) =2

a(α) + b(α)

[f

(x

a(α)

)Ix ge 0 + f

(x

b(α)

)Ix lt 0

] x isin IR (30)






5 DISCUSSION



APPENDIX


MX(s) = E[exps⊤X]

=


VΩV)


(x minus ξV

) + ξU


∆)

Φq(C ξU

ΩU)dx

= exp

s⊤ξ

V+

1

2s⊤ΩVs

1

Φq(C ξU

ΩU)

times

intφp(x ξ


V(x minus ξ

V) + ξ


V∆) dx

= exp

s⊤ξ

V+

1

2s⊤ΩVs

1

Φq(C ξU




V(xminusξ

V)+ξ




⊤s+ξU



ΩU) 2


U)) whereW sim G



(radicW s

)]

= exps⊤ξV

intinfin

0exp1

2ws⊤ΩVsΦq


)dG(w)

Φq(00ΩU)

= exps⊤ξV

int infin

0

exp

w

2s⊤ΩVs

dG(w)

Φq(∆⊤s0ΩU)

Φq(00ΩU)


⊤s0ΩU)

Φq(00ΩU)

2


REFERENCES







































































Santiago 22 Chile










SLCT

COther subsetsβ

FUSE

FSS

SUN

CFUSNCSN

SMSN

SESUEFUSN

SE

GSE

SN

GSNFSN FSE

SS

HSN

q=1

FUSS C( )

etc

C( minus )

C( ) C(0)

α

α

β

α

SMSS














FUSS -symmetric

























U= a ξ




YΩY)

Φr(C(β) ξY





Φq(aβ AA⊤)















P(U gt 0) (28)


fX(x) = 2qfV(x)FT(w(x)) (29)


intinfin










fX(x) =2

a(α) + b(α)

[f

(x

a(α)

)Ix ge 0 + f

(x

b(α)

)Ix lt 0

] x isin IR (30)






5 DISCUSSION



APPENDIX


MX(s) = E[exps⊤X]

=


VΩV)


(x minus ξV

) + ξU


∆)

Φq(C ξU

ΩU)dx

= exp

s⊤ξ

V+

1

2s⊤ΩVs

1

Φq(C ξU

ΩU)

times

intφp(x ξ


V(x minus ξ

V) + ξ


V∆) dx

= exp

s⊤ξ

V+

1

2s⊤ΩVs

1

Φq(C ξU




V(xminusξ

V)+ξ




⊤s+ξU



ΩU) 2


U)) whereW sim G



(radicW s

)]

= exps⊤ξV

intinfin

0exp1

2ws⊤ΩVsΦq


)dG(w)

Φq(00ΩU)

= exps⊤ξV

int infin

0

exp

w

2s⊤ΩVs

dG(w)

Φq(∆⊤s0ΩU)

Φq(00ΩU)


⊤s0ΩU)

Φq(00ΩU)

2


REFERENCES







































































Santiago 22 Chile


















FUSS -symmetric

























U= a ξ




YΩY)

Φr(C(β) ξY





Φq(aβ AA⊤)















P(U gt 0) (28)


fX(x) = 2qfV(x)FT(w(x)) (29)


intinfin










fX(x) =2

a(α) + b(α)

[f

(x

a(α)

)Ix ge 0 + f

(x

b(α)

)Ix lt 0

] x isin IR (30)






5 DISCUSSION



APPENDIX


MX(s) = E[exps⊤X]

=


VΩV)


(x minus ξV

) + ξU


∆)

Φq(C ξU

ΩU)dx

= exp

s⊤ξ

V+

1

2s⊤ΩVs

1

Φq(C ξU

ΩU)

times

intφp(x ξ


V(x minus ξ

V) + ξ


V∆) dx

= exp

s⊤ξ

V+

1

2s⊤ΩVs

1

Φq(C ξU




V(xminusξ

V)+ξ




⊤s+ξU



ΩU) 2


U)) whereW sim G



(radicW s

)]

= exps⊤ξV

intinfin

0exp1

2ws⊤ΩVsΦq


)dG(w)

Φq(00ΩU)

= exps⊤ξV

int infin

0

exp

w

2s⊤ΩVs

dG(w)

Φq(∆⊤s0ΩU)

Φq(00ΩU)


⊤s0ΩU)

Φq(00ΩU)

2


REFERENCES







































































Santiago 22 Chile













U= a ξ




YΩY)

Φr(C(β) ξY





Φq(aβ AA⊤)















P(U gt 0) (28)


fX(x) = 2qfV(x)FT(w(x)) (29)


intinfin










fX(x) =2

a(α) + b(α)

[f

(x

a(α)

)Ix ge 0 + f

(x

b(α)

)Ix lt 0

] x isin IR (30)






5 DISCUSSION



APPENDIX


MX(s) = E[exps⊤X]

=


VΩV)


(x minus ξV

) + ξU


∆)

Φq(C ξU

ΩU)dx

= exp

s⊤ξ

V+

1

2s⊤ΩVs

1

Φq(C ξU

ΩU)

times

intφp(x ξ


V(x minus ξ

V) + ξ


V∆) dx

= exp

s⊤ξ

V+

1

2s⊤ΩVs

1

Φq(C ξU




V(xminusξ

V)+ξ




⊤s+ξU



ΩU) 2


U)) whereW sim G



(radicW s

)]

= exps⊤ξV

intinfin

0exp1

2ws⊤ΩVsΦq


)dG(w)

Φq(00ΩU)

= exps⊤ξV

int infin

0

exp

w

2s⊤ΩVs

dG(w)

Φq(∆⊤s0ΩU)

Φq(00ΩU)


⊤s0ΩU)

Φq(00ΩU)

2


REFERENCES







































































Santiago 22 Chile













P(U gt 0) (28)


fX(x) = 2qfV(x)FT(w(x)) (29)


intinfin










fX(x) =2

a(α) + b(α)

[f

(x

a(α)

)Ix ge 0 + f

(x

b(α)

)Ix lt 0

] x isin IR (30)






5 DISCUSSION



APPENDIX


MX(s) = E[exps⊤X]

=


VΩV)


(x minus ξV

) + ξU


∆)

Φq(C ξU

ΩU)dx

= exp

s⊤ξ

V+

1

2s⊤ΩVs

1

Φq(C ξU

ΩU)

times

intφp(x ξ


V(x minus ξ

V) + ξ


V∆) dx

= exp

s⊤ξ

V+

1

2s⊤ΩVs

1

Φq(C ξU




V(xminusξ

V)+ξ




⊤s+ξU



ΩU) 2


U)) whereW sim G



(radicW s

)]

= exps⊤ξV

intinfin

0exp1

2ws⊤ΩVsΦq


)dG(w)

Φq(00ΩU)

= exps⊤ξV

int infin

0

exp

w

2s⊤ΩVs

dG(w)

Φq(∆⊤s0ΩU)

Φq(00ΩU)


⊤s0ΩU)

Φq(00ΩU)

2


REFERENCES







































































Santiago 22 Chile













5 DISCUSSION



APPENDIX


MX(s) = E[exps⊤X]

=


VΩV)


(x minus ξV

) + ξU


∆)

Φq(C ξU

ΩU)dx

= exp

s⊤ξ

V+

1

2s⊤ΩVs

1

Φq(C ξU

ΩU)

times

intφp(x ξ


V(x minus ξ

V) + ξ


V∆) dx

= exp

s⊤ξ

V+

1

2s⊤ΩVs

1

Φq(C ξU




V(xminusξ

V)+ξ




⊤s+ξU



ΩU) 2


U)) whereW sim G



(radicW s

)]

= exps⊤ξV

intinfin

0exp1

2ws⊤ΩVsΦq


)dG(w)

Φq(00ΩU)

= exps⊤ξV

int infin

0

exp

w

2s⊤ΩVs

dG(w)

Φq(∆⊤s0ΩU)

Φq(00ΩU)


⊤s0ΩU)

Φq(00ΩU)

2


REFERENCES







































































Santiago 22 Chile











V(xminusξ

V)+ξ




⊤s+ξU



ΩU) 2


U)) whereW sim G



(radicW s

)]

= exps⊤ξV

intinfin

0exp1

2ws⊤ΩVsΦq


)dG(w)

Φq(00ΩU)

= exps⊤ξV

int infin

0

exp

w

2s⊤ΩVs

dG(w)

Φq(∆⊤s0ΩU)

Φq(00ΩU)


⊤s0ΩU)

Φq(00ΩU)

2


REFERENCES







































































Santiago 22 Chile



































































Santiago 22 Chile









A unied view on skewed distributions arising from selections

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Transcript of A unied view on skewed distributions arising from selections