The gamlss.dist Package -...

The gamlss.dist PackageOctober 3, 2007

Type Package

Title Extra distributions to be used for GAMLSS modelling.

Version 1.7-0

Date 2007-10-01

Author Mikis Stasinopoulos <[email protected]>, Bob Rigby<[email protected]> with contributions from Calliope Akantziliotou, Raydonal Ospina<[email protected]> and Nicoletta Motpan.

Maintainer Mikis Stasinopoulos <[email protected]>

Depends R (>= 2.4.0), gamlss

Description This package contains few extra distributions for GAMLSS modelling.

License GPL-2 | GPL-3

URL http://www.gamlss.com/

R topics documented:BEOI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2BEZI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4DEL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7EXP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9GG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10GIG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12NOF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14RGE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16SHASH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18SICHEL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21ST3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23WEI3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25ZIP2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27exGAUS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28gamlss.dist-package . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30parzen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

1

2 BEOI

Index 35

BEOI The one-inflated beta distribution for fitting a GAMLSS

Description

The function BEOI() defines the one-inflated beta distribution, a three parameter distribution, fora gamlss.family object to be used in GAMLSS fitting using the function gamlss(). Theone-inflated beta is similar to the beta distribution but allows ones as y values. This distributionis an extension of the beta distribution using a parameterization of the beta law that is indexedby mean and precision parameters (Ferrari and Cribari-Neto, 2004). The extra parameter modelsthe probability at one. The functions dBEOI, pBEOI, qBEOI and rBEOI define the density,distribution function, quantile function and random generation for the BEOI parameterization ofthe one-inflated beta distribution. plotBEOI can be used to plot the distribution. meanBEOIcalculates the expected value of the response for a fitted model.

Usage

BEOI(mu.link = "logit", sigma.link = "log", nu.link = "logit")

dBEOI(y, mu = 0.5, sigma = 1, nu = 0.1, log = FALSE)

pBEOI(q, mu = 0.5, sigma = 1, nu = 0.1, lower.tail = TRUE, log.p = FALSE)

qBEOI(p, mu = 0.5, sigma = 1, nu = 0.1, lower.tail = TRUE,log.p = FALSE)

rBEOI(n, mu = 0.5, sigma = 1, nu = 0.1)

plotBEOI(mu = .5, sigma = 1, nu = 0.1, from = 0.001, to = 1, n = 101,...)

meanBEOI(obj)

Arguments

mu.link the mu link function with default logit

sigma.link the sigma link function with default log

nu.link the nu link function with default logit

y,q vector of quantiles

mu vector of location parameter values

sigma vector of precision parameter values

nu vector of parameter values modelling the probability at one

log, log.p logical; if TRUE, probabilities p are given as log(p).

lower.tail logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x]

BEOI 3

p vector of probabilities.

n number of observations. If length(n) > 1, the length is taken to be thenumber required

from where to start plotting the distribution from

to up to where to plot the distribution

obj a fitted BEOI object

... other graphical parameters for plotting

Details

The one-inflated beta distribution is given as

f(y) = ν

if (y = 1)

f(y|µ, σ) = (1− ν)Γ(σ)

Γ(µσ)Γ((1− µ)σ)yµσ(1− y)((1−µ)σ)−1

if y = (0, 1). The parameters satisfy 0 < µ < 0, σ > 0 and 0 < ν < 1.

Here E(y) = ν + (1− ν)µ and V ar(y) = (1− ν)µ(1−µ)σ+1 + ν(1− ν)(1− µ)2.

Value

returns a gamlss.family object which can be used to fit a one-inflated beta distribution in thegamlss() function.

Note

This work is part of my PhD project at the University of Sao Paulo under the supervion of ProfessorSilvia Ferrari. My thesis is concerned with regression modelling of rates and proportions withexcess of zeros and/or ones

Author(s)

Raydonal Ospina, Department of Statistics, University of Sao Paulo, Brazil.

〈[email protected]〉

References

Ferrari, S.L.P., Cribari-Neto, F. (2004). Beta regression for modelling rates and proportions. Jour-nal of Applied Statistics, 31 (1), 799-815.

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale andshape (with discussion). Applied Statistics, 54 (3), 507-554.

Stasinopoulos D. M., Rigby R.A. and Akantziliotou C. (2006). Instructions on how to use theGAMLSS package in R. Accompanying documentation in the current GAMLSS help files (see alsohttp://www.gamlss.com/).

http://www.gamlss.com/

4 BEZI

See Also

gamlss, gamlss.family, BEOI

Examples

BEOI()# gives information about the default links for the BEOI distribution# plotting the distributionplotBEOI( mu =0.5 , sigma=5, nu = 0.1, from = 0.001, to=1, n = 101)# plotting the cdfplot(function(y) pBEOI(y, mu=.5 ,sigma=5, nu=0.1), 0.001, 0.999)# plotting the inverse cdfplot(function(y) qBEOI(y, mu=.5 ,sigma=5, nu=0.1), 0.001, 0.999)# generate random numbersdat<-rBEOI(100, mu=.5, sigma=5, nu=0.1)# fit a model to the data.mod1<-gamlss(dat~1,sigma.formula=~1, nu.formula=~1, family=BEOI)fitted(mod1)[1]summary(mod1)fitted(mod1,"mu")[1] #fitted mufitted(mod1,"sigma")[1] #fitted sigmafitted(mod1,"nu")[1] #fitted numeanBEOI(mod1)[1] # expected value of the response

BEZI The zero-inflated beta distribution for fitting a GAMLSS

Description

The function BEZI() defines the zero-inflated beta distribution, a three parameter distribution, fora gamlss.family object to be used in GAMLSS fitting using the function gamlss(). Thezero-inflated beta is similar to the beta distribution but allows zeros as y values. This distributionis an extension of the beta distribution using a parameterization of the beta law that is indexedby mean and precision parameters (Ferrari and Cribari-Neto, 2004). The extra parameter modelsthe probability at zero. The functions dBEZI, pBEZI, qBEZI and rBEZI define the density,distribution function, quantile function and random generation for the BEZI parameterization ofthe zero-inflated beta distribution. plotBEZI can be used to plot the distribution. meanBEZIcalculates the expected value of the response for a fitted model.

Usage

BEZI(mu.link = "logit", sigma.link = "log", nu.link = "logit")

dBEZI(y, mu = 0.5, sigma = 1, nu = 0.1, log = FALSE)

pBEZI(q, mu = 0.5, sigma = 1, nu = 0.1, lower.tail = TRUE, log.p = FALSE)

qBEZI(p, mu = 0.5, sigma = 1, nu = 0.1, lower.tail = TRUE,

BEZI 5

log.p = FALSE)

rBEZI(n, mu = 0.5, sigma = 1, nu = 0.1)

plotBEZI(mu = .5, sigma = 1, nu = 0.1, from = 0, to = 0.999, n = 101,...)

meanBEZI(obj)

Arguments

mu.link the mu link function with default logit

sigma.link the sigma link function with default log

nu.link the nu link function with default logit



sigma vector of precision parameter values

nu vector of parameter values modelling the probability at zero





from where to start plotting the distribution from

to up to where to plot the distribution

obj a fitted BEZI object

... other graphical parameters for plotting

Details

The zero-inflated beta distribution is given as

f(y) = ν

if (y = 0)

f(y|µ, σ) = (1− ν)Γ(σ)

Γ(µσ)Γ((1− µ)σ)yµσ(1− y)((1−µ)σ)−1

if y = (0, 1). The parameters satisfy 0 < µ < 0, σ > 0 and 0 < ν < 1.

Here E(y) = (1− ν)µ and V ar(y) = (1− ν)µ(1−µ)σ+1 + ν(1− ν)µ2.

Value

returns a gamlss.family object which can be used to fit a zero-inflated beta distribution in thegamlss() function.

6 BEZI

Note

This work is part of my PhD project at the University of Sao Paulo under the supervion of ProfessorSilvia Ferrari. My thesis is concerned with regression modelling of rates and proportions withexcess of zeros and/or ones

Author(s)

Raydonal Ospina, Department of Statistics, University of Sao Paulo, Brazil.

〈[email protected]〉

References

Ferrari, S.L.P., Cribari-Neto, F. (2004). Beta regression for modelling rates and proportions. Jour-nal of Applied Statistics, 31 (1), 799-815.

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale andshape (with discussion). Applied Statistics, 54 (3), 507-554.

Stasinopoulos D. M., Rigby R.A. and Akantziliotou C. (2006). Instructions on how to use theGAMLSS package in R. Accompanying documentation in the current GAMLSS help files (see alsohttp://www.gamlss.com/).

See Also

gamlss, gamlss.family, BEZI

Examples

BEZI()# gives information about the default links for the BEZI distribution# plotting the distributionplotBEZI( mu =0.5 , sigma=5, nu = 0.1, from = 0, to=0.99, n = 101)# plotting the cdfplot(function(y) pBEZI(y, mu=.5 ,sigma=5, nu=0.1), 0, 0.999)# plotting the inverse cdfplot(function(y) qBEZI(y, mu=.5 ,sigma=5, nu=0.1), 0, 0.999)# generate random numbersdat<-rBEZI(100, mu=.5, sigma=5, nu=0.1)# fit a model to the data. Tits a constant for mu, sigma and numod1<-gamlss(dat~1,sigma.formula=~1, nu.formula=~1, family=BEZI)fitted(mod1)[1]summary(mod1)fitted(mod1,"mu")[1] #fitted mufitted(mod1,"sigma")[1] #fitted sigmafitted(mod1,"nu")[1] #fitted numeanBEZI(mod1)[1] # expected value of the response


DEL 7

DEL The Delaporte distribution for fitting a GAMLSS model

Description

The DEL() function defines the Delaporte distribution, a three parameter discrete distribution, fora gamlss.family object to be used in GAMLSS fitting using the function gamlss(). Thefunctions dDEL, pDEL, qDEL and rDEL define the density, distribution function, quantile functionand random generation for the Delaporte DEL(), distribution.

Usage

DEL(mu.link = "log", sigma.link = "log", nu.link = "logit")dDEL(y, mu=1, sigma=1, nu=0.5, log=FALSE)pDEL(q, mu=1, sigma=1, nu=0.5, lower.tail = TRUE,

log.p = FALSE)qDEL(p, mu=1, sigma=1, nu=0.5, lower.tail = TRUE,

log.p = FALSE, max.value = 10000)rDEL(n, mu=1, sigma=1, nu=0.5, max.value = 10000)

Arguments

mu.link Defines the mu.link, with "log" link as the default for the mu parameter

sigma.link Defines the sigma.link, with "log" link as the default for the sigma parame-ter

nu.link Defines the nu.link, with "logit" link as the default for the nu parameter

y vector of (non-negative integer) quantiles

mu vector of positive mu

sigma vector of positive dispersion parameter

nu vector of nu

p vector of probabilities

q vector of quantiles

n number of random values to return

log, log.p logical; if TRUE, probabilities p are given as log(p)


max.value a constant, set to the default value of 10000 for how far the algorithm shouldlook for q

8 DEL

Details

The probability function of the Delaporte distribution is given by

f(y|µ, σ, ν) =e−µν

Γ(1/σ)[1 + µσ(1− ν)]−1/σ

S

where

S =y∑

j=0

( yj )µyνy−j

y!

[µ +

1σ(1− ν)

]−j

Γ(

1σ

+ j

)for y = 0, 1, 2, ...,∞ where µ > 0 , σ > 0 and 0 < ν < 1. This distribution is a parametrizationof the distribution given by Wimmer and Altmann (1999) p 515-516 where α = µν, k = 1/σ andρ = [1 + µσ(1− ν)]−1

Value

Returns a gamlss.family object which can be used to fit a Delaporte distribution in the gamlss()function.

Note

The mean of Y is given by E(Y ) = µ and the variance by V (Y ) = µ + µ2σ (1− ν)2.

Author(s)

Rigby, R. A. and Stasinopoulos D. M.

References

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale andshape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R. A., Stasinopoulos D. M. and Akantziliotou, C. (2006) Modelling the parameters of a fam-ily of mixed Poisson distributions including the Sichel and Delaptorte. Submitted for publication.

Stasinopoulos D. M., Rigby R.A. and Akantziliotou C. (2003) Instructions on how to use theGAMLSS package in R. Accompanying documentation in the current GAMLSS help files, (seealso http://www.gamlss.com/).

Wimmer, G. and Altmann, G (1999). Thesaurus of univariate discrete probability distributions .Stamn Verlag, Essen, Germany

See Also

gamlss, gamlss.family, SI , SICHEL

Examples

DEL()# gives information about the default links for the Delaporte distribution#plot the pdf using plotplot(function(y) dDEL(y, mu=10, sigma=1, nu=.5), from=0, to=100, n=100+1, type="h") # pdf# plot the cdf


EXP 9

plot(seq(from=0,to=100),pDEL(seq(from=0,to=100), mu=10, sigma=1, nu=0.5), type="h") # cdf# generate random sampletN <- table(Ni <- rDEL(100, mu=10, sigma=1, nu=0.5))r <- barplot(tN, col='lightblue')# fit a model to the datagamlss(Ni~1,family=DEL, control=gamlss.control(n.cyc=50))

EXP Exponential distribution for fitting a GAMLSS

Description

The function EXP defines the exponential distribution, a one parameter distribution for a gamlss.familyobject to be used in GAMLSS fitting using the function gamlss(). The mu parameter representsthe mean of the distribution. The functions dEXP, pEXP, qEXP and rEXP define the density, dis-tribution function, quantile function and random generation for the specific parameterization of theexponential distribution defined by function EXP.

Usage

EXP(mu.link ="log")dEXP(y, mu = 1, log = FALSE)pEXP(q, mu = 1, lower.tail = TRUE, log.p = FALSE)qEXP(p, mu = 1, lower.tail = TRUE, log.p = FALSE)rEXP(n, mu = 1)

Arguments

mu.link Defines the mu.link, with "log" link as the default for the mu parameter, otherlinks are "inverse" and "identity"






n number of observations. If length(n) > 1, the length is taken to be the numberrequired

Details

The specific parameterization of the exponential distribution used in EXP is

f(y|µ) =1µ

exp{− y

µ

}, for y>0, µ > 0.

10 GG

Value

EXP() returns a gamlss.family object which can be used to fit an exponential distribution in thegamlss() function. dEXP() gives the density, pEXP() gives the distribution function, qEXP() givesthe quantile function, and rEXP() generates random deviates.

Note

Author(s)

Mikis Stasinopoulos 〈[email protected]〉, Bob Rigby 〈[email protected]〉and Nicoleta Motpan

References



See Also

gamlss, gamlss.family

Examples

library(gamlss)y<-rEXP(1000,mu=1) # generates 1000 random observationshistDist(y, family=EXP)

GG Generalized Gamma distribution for fitting a GAMLSS

Description

The function GG defines the generalized gamma distribution, a three parameter distribution, for agamlss.family object to be used in GAMLSS fitting using the function gamlss(). The parameteri-zation used has the mean of the distribution equal to mu and the variance equal to (σ2)(µ2). Thefunctions dGG, pGG, qGG and rGG define the density, distribution function, quantile function andrandom generation for the specific parameterization of the generalized gamma distribution definedby function GG.


GG 11

Usage

GG(mu.link = "log", sigma.link = "log",nu.link = "identity")

dGG(y, mu=1, sigma=0.1, nu=1,log = FALSE)

pGG(q, mu=1, sigma=0.1, nu=1, lower.tail = TRUE,log.p = FALSE)

qGG(p, mu=1, sigma=0.1, nu=1, lower.tail = TRUE,log.p = FALSE )

rGG(n, mu=1, sigma=0.1, nu=1)

Arguments


sigma.link Defines the sigma.link, with "log" link as the default for the sigma parameter,other links are "inverse" and "identity"

nu.link Defines the nu.link, with "identity" link as the default for the sigma parameter,other links are 1/nu2 and "log"



sigma vector of scale parameter values

nu vector of shape parameter values





Details

The specific parameterization of the generalized gamma distribution used in GG is

f(y|µ, σ, ν) =θθzθνe( − θz)

(Γ(θ)y)

where z = (y/µ)ν , θ = 1/(σ2|ν|2) for y>0, µ > 0, σ > 0 and −∞ < ν < +∞. Note that forν = 0 the distribution is log normal.

Value

GG() returns a gamlss.family object which can be used to fit a generalized gamma distribution inthe gamlss() function. dGG() gives the density, pGG() gives the distribution function, qGG() givesthe quantile function, and rGG() generates random deviates.

12 GIG

Note

Author(s)

Mikis Stasinopoulos [email protected], Bob Rigby [email protected] andNicoleta Motpan

References

Lopatatzidis, A. and Green, P. J. (2000), Nonparametric quantile regression using the gamma dis-tribution, unpublished.



See Also

gamlss, gamlss.family, GA

Examples

y<-rGG(100,mu=1,sigma=0.1, nu=-.5) # generates 100 random observationshistDist(y, family=GG)m1 <-gamlss(y~1,family=GG)prof.dev(m1, "nu", min=-2, max=2, step=0.2)

GIG Generalized Inverse Gaussian distribution for fitting a GAMLSS

Description

The function GIG defines the generalized inverse gaussian distribution, a three parameter distribu-tion, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss().The functions dGIG, pGIG, qGIG and rGIG define the density, distribution function, quantilefunction and random generation for the specific parameterization of the generalized inverse gaus-sian distribution defined by function GIG.

Usage

GIG(mu.link = "log", sigma.link = "log",nu.link = "identity")

dGIG(y, mu=1, sigma=1, nu=1,log = FALSE)

pGIG(q, mu=1, sigma=1, nu=1, lower.tail = TRUE,


GIG 13

log.p = FALSE)qGIG(p, mu=1, sigma=1, nu=1, lower.tail = TRUE,

log.p = FALSE,lower.limit = 0,upper.limit = mu+10*sqrt(sigma^2*mu^3))

rGIG(n, mu=1, sigma=1, nu=1, ...)

Arguments


sigma.link Defines the sigma.link, with "log" link as the default for the sigma parameter,other links are "inverse" and "identity"

nu.link Defines the nu.link, with "identity" link as the default for the nu parameter, otherlinks are "inverse" and "log"




nu vector of shape parameter values





lower.limit a constant, set to the value of 0 for the algorithm to know from where to beginlooking for q

upper.limit a constant, set to the value of mu+10*sqrt(sigma^2*mu^3) for how farthe algorithm should look for q

... for extra arguments

Details

The specific parameterization of the generalized inverse gaussian distribution used in GIG is f(y|µ, σ, ν) =( c

µ )ν( y(ν−1)

2K( 1σ ,ν)

)(exp((−12σ )( cy

µ + µcy ))) where c = K( 1

σ ,ν+1)

K( 1σ ,ν)

, for y>0, µ > 0, σ > 0 and −∞ <

ν < +∞.

Value

GIG() returns a gamlss.family object which can be used to fit a generalized inverse gaussian distri-bution in the gamlss() function. dGIG() gives the density, pGIG() gives the distribution function,qGIG() gives the quantile function, and rGIG() generates random deviates.

Note

14 NOF

Author(s)

Mikis Stasinopoulos 〈[email protected]〉, Bob Rigby 〈[email protected]〉and Nicoleta Motpan

References



Jorgensen B. (1982) Statistical properties of the generalized inverse Gaussian distribution, Series:Lecture notes in statistics; 9, New York : Springer-Verlag.

See Also

gamlss, gamlss.family, GI

Examples

y<-rGIG(100,mu=1,sigma=1, nu=-0.5) # generates 1000 random observationshistDist(y, family=GIG)

NOF Normal distribution family for fitting a GAMLSS

Description

The function NOF() defines a normal distribution family, which has three parameters. The distri-bution can be used in a GAMLSS fitting using the function gamlss(). The mean of NOF is equalto mu. The variance is equal to sigmatextasciicircum{}2*mutextasciicircum{}nu so the standard deviation is sigma*mutextasciicircum{}(nu/2). The function is design for cases where the variance is propor-tional to a power of the mean. The functions dNOF, pNOF, qNOF and rNOF define the density,distribution function, quantile function and random generation for the NOF parametrization of thenormal distribution family.

Usage

NOF(mu.link = "identity", sigma.link = "log", nu.link = "identity")dNOF(y, mu = 0, sigma = 1, nu = 0, log = FALSE)pNOF(q, mu = 0, sigma = 1, nu = 0, lower.tail = TRUE, log.p = FALSE)qNOF(p, mu = 0, sigma = 1, nu = 0, lower.tail = TRUE, log.p = FALSE)rNOF(n, mu = 0, sigma = 1, nu = 0)


NOF 15

Arguments

mu.link Defines the mu.link, with "identity" link as the default for the mu parameter

sigma.link Defines the sigma.link, with "log" link as the default for the sigma parame-ter

nu.link Defines the nu.link with "identity" link as the default for the nu parameter




nu vector of power parameter values





Details

The parametrization of the normal distribution given in the function NOF() is

f(y|µ, σ, ν) =1√

2πσµν/2exp

[−1

2(y − µ)2

σ2µν

]

for y = (−∞,∞), µ = (−∞,∞), σ > 0 and ν = (−∞,+∞).

Value

returns a gamlss.family object which can be used to fit a normal distribution family in thegamlss() function.

Note

For the function NOF(), µ is the mean and σµν/2 is the standard deviation of the normal distributionfamily. The NOF is design for fitting regression type models where the variance is proportional to apower ofthe mean. Models of this type are related to the "pseudo likelihood" models of Carroll andRubert (1987) but here a proper likelihood is miximised.

Note that because the high correlation between the sigma and the nu parameter the mixed()method should be used in the fitting.

Author(s)

Mikis Stasinopoulos, Bob Rigby and Calliope Akantziliotou

16 RGE

References

Davidian, M. and Carroll, R. J. (1987), Variance Function Estimation, Journal of the AmericanStatistical Association, Vol. 82, pp. 1079-1091



See Also

gamlss, gamlss.family, NO

Examples

NOF()# gives information about the default links for the normal distribution familydata(abdom)# the normal distribution fit with constant sigmam1<-gamlss(y~poly(x,2), sigma.fo=~1, family=NO, data=abdom)# the normal family fit with variance proportional to mum2<-gamlss(y~poly(x,2), sigma.fo=~1, family=NOF, data=abdom, method=mixed(1,20))# a nornal distribution fit with variance as a function of xm3 <-gamlss(y~poly(x,2), sigma.fo=~x, family=NO, data=abdom, method=mixed(1,20))GAIC(m1,m2,m3)

RGE Reverse generalized extreme family distribution for fitting a GAMLSS

Description

The function RGE defines the reverse generalized extreme family distribution, a three parameterdistribution, for a gamlss.family object to be used in GAMLSS fitting using the functiongamlss(). The functions dRGE, pRGE, qRGE and rRGE define the density, distribution function,quantile function and random generation for the specific parameterization of the reverse generalizedextreme distribution given in details below.

Usage

RGE(mu.link = "identity", sigma.link = "log", nu.link = "log")dRGE(y, mu = 1, sigma = 0.1, nu = 1, log = FALSE)pRGE(q, mu = 1, sigma = 0.1, nu = 1, lower.tail = TRUE, log.p = FALSE)qRGE(p, mu = 1, sigma = 0.1, nu = 1, lower.tail = TRUE, log.p = FALSE)rRGE(n, mu = 1, sigma = 0.1, nu = 1)


RGE 17

Arguments

mu.link Defines the mu.link, with "identity" link as the default for the mu parametersigma.link Defines the sigma.link, with "log" link as the default for the sigma parame-

ternu.link Defines the nu.link, with "log" link as the default for the nu parametery,q vector of quantilesmu vector of location parameter valuessigma vector of scale parameter valuesnu vector of the shape parameter valueslog, log.p logical; if TRUE, probabilities p are given as log(p).lower.tail logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x]p vector of probabilities.n number of observations. If length(n) > 1, the length is taken to be the

number required

Details

Definition file for reverse generalized extreme family distribution.

The probability density function of the generalized extreme value distribution is obtained fromJohnson et al. (1995), Volume 2, p76, equation (22.184) [where (ξ, θ, γ) −→ (µ, σ, ν)].

The probability density function of the reverse generalized extreme value distribution is then ob-tained by replacing y by -y and µ by −µ.

Hence the probability density function of the reverse generalized extreme value distribution withν > 0 is given by

f(y|µ, σ, ν) =1σ

[1 +

ν(y − µ)σ

] 1ν−1

S1(y|µ, σ, ν)

forµ− σ

ν< y < ∞

where

S1(y|µ, σ, ν) = exp

{−

[1 +

ν(y − µ)σ

] 1ν

}

and where −∞ < µ < y + σν , σ > 0 and ν > 0. Note that only the case ν > 0 is allowed

here. The reverse generalized extreme value distribution is denoted as RGE(µ, σ, ν) or as ReverseGeneralized.Extreme. Family(µ, σ, ν).

Note the the above distribution is a reparameterization of the three parameter Weibull distributiongiven by

f(y|α1, α2, α3) =α3

α2

[y − α1

α2

]α3−1

exp[−

(y − α1

α2

)α3]

given by setting α1 = µ− σ/ν, α2 = σ/ν, α3 = 1/ν.

18 SHASH

Value

RGE() returns a gamlss.family object which can be used to fit a reverse generalized extremedistribution in the gamlss() function. dRGE() gives the density, pRGE() gives the distributionfunction, qRGE() gives the quantile function, and rRGE() generates random deviates.

Note

Author(s)

Bob Rigby 〈[email protected]〉, Mikis Stasinopoulos 〈[email protected]〉and Kalliope Akantziliotou

References



See Also


Examples

RGE()# gives information about the default links for the reverse generalized extreme family distributionnewdata<-rRGE(1000,mu=0,sigma=1,nu=5) # generates 1000 random observationshistDist(newdata)rm(h,newdata)

SHASH The Sinh-Arcsinh (SHASH) distribution for fitting a GAMLSS

Description

This function defines the Sinh-Arcsinh (SHASH) distribution, a four parameter distribution, for agamlss.family object to be used for a GAMLSS fitting using the function gamlss(). Thefunctions dSHASH, pSHASH, qSHASH and rSHASH define the density, distribution function, quan-tile function and random generation for the Sinh-Arcsinh (SHASH) distribution.


SHASH 19

Usage

SHASH(mu.link = "identity", sigma.link = "log", nu.link = "log",tau.link = "log")

dSHASH(y, mu = 0, sigma = 1, nu = 1, tau = .5, log = FALSE)pSHASH(q, mu = 0, sigma = 1, nu = 1, tau = .5, lower.tail = TRUE,

log.p = FALSE)qSHASH(p, mu = 0, sigma = 1, nu = 1, tau = .5, lower.tail = TRUE,

log.p = FALSE, lower.limit = mu-10*(sigma/(nu*tau)),upper.limit = mu+10*(sigma/(nu*tau)))

rSHASH(n, mu = 0, sigma = 1, nu = 1, tau = .5)

Arguments

mu.link Defines the mu.link, with "identity" link as the default for the mu parameter.Other links are "1/mu2" and "log"

sigma.link Defines the sigma.link, with "log" link as the default for the sigma param-eter. Other links are "inverse" and "identity"

nu.link Defines the nu.link, with "log" link as the default for the nu parameter. Otherlinks are "1/nu2" and "log"

tau.link Defines the tau.link, with "log" link as the default for the tau parameter.Other links are "1/tau2", and "identity




nu vector of skewness nu parameter values

tau vector of kurtosis tau parameter values





lower.limit lower limit for the golden search to find quantiles from probabilities

upper.limit upper limit for the golden search to find quantiles from probabilities

Details

The probability density function of the Sinh-Arcsinh distribution, (SHASH), Jones(2005), is definedas

f(y|µ, σ ν, τ) =c√

2πσ(1 + z2)1/2e−r2/2

where

20 SHASH

r =12

{exp

[τ sinh−1(z)

]− exp

[−ν sinh−1(z)

]}and

c =12

{τ exp

[τ sinh−1(z)

]+ ν exp

[−ν sinh−1(z)

]}and z = (y − µ)/σ for −∞ < y < ∞, µ = (−∞,+∞), σ > 0, ν > 0) and τ > 0.

The parameters µ and σ are the location and scale of the distribution. The parameter ν determinesthe left hand tail of the distribution with ν > 1 indicating a lighter tail than the normal and ν < 1heavier tail than the normal. The parameter τ determines the right hand tail of the distribution inthe same way.

Value

SHASH() returns a gamlss.family object which can be used to fit the SHASH distribution inthe gamlss() function. dSHASH() gives the density, pSHASH() gives the distribution function,qSHASH() gives the quantile function, and rSHASH() generates random deviates.

Warning

The qSHASH and rSHASH are slow since they are relying on golden section for finding the quan-tiles

Note

Author(s)

Bob Rigby 〈[email protected]〉 and Mikis Stasinopoulos 〈[email protected]〉

References

Jones, M. C. (2006) p 546-547 in the discussion of Rigby, R. A. and Stasinopoulos D. M. (2005)Appl. Statist., 54, part 3.


Stasinopoulos D. M. Rigby R. A. and Akantziliotou C. (2006) Instructions on how to use theGAMLSS package in R. Accompanying documentation in the current GAMLSS help files, (seealso http://www.gamlss.com/).

See Also

gamlss, gamlss.family, JSU, BCT


SICHEL 21

Examples

SHASH() #plot(function(x)dSHASH(x, mu=0,sigma=1, nu=1, tau=2), -5, 5,main = "The SHASH density mu=0,sigma=1,nu=1, tau=2")plot(function(x) pSHASH(x, mu=0,sigma=1,nu=1, tau=2), -5, 5,main = "The BCPE cdf mu=0, sigma=1, nu=1, tau=2")dat<-rSHASH(100,mu=10,sigma=1,nu=1,tau=1.5)gamlss(dat~1,family=SHASH, control=gamlss.control(n.cyc=30))

SICHEL The Sichel dustribution for fitting a GAMLSS model

Description

The SICHEL() function defines the Sichel distribution, a three parameter discrete distribution,for a gamlss.family object to be used in GAMLSS fitting using the function gamlss(). Thefunctions dSICHEL, pSICHEL, qSICHEL and rSICHEL define the density, distribution function,quantile function and random generation for the Sichel SICHEL(), distribution.

Usage

SICHEL(mu.link = "log", sigma.link = "log", nu.link = "identity")dSICHEL(y, mu=1, sigma=1, nu=-0.5, log=FALSE)pSICHEL(q, mu=1, sigma=1, nu=-0.5, lower.tail = TRUE,

log.p = FALSE)qSICHEL(p, mu=1, sigma=1, nu=-0.5, lower.tail = TRUE,

log.p = FALSE, max.value = 10000)rSICHEL(n, mu=1, sigma=1, nu=-0.5, max.value = 10000)

Arguments

mu.link Defines the mu.link, with "log" link as the default for the mu parametersigma.link Defines the sigma.link, with "log" link as the default for the sigma parame-

ternu.link Defines the nu.link, with "identity" link as the default for the nu parametery vector of (non-negative integer) quantilesmu vector of positive musigma vector of positive despersion parameternu vector of nup vector of probabilitiesq vector of quantilesn number of random values to returnlog, log.p logical; if TRUE, probabilities p are given as log(p)lower.tail logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x]max.value a constant, set to the default value of 10000 for how far the algorithm should

look for q

22 SICHEL

Details

The probability function of the Sichel distribution is given by

f(y|µ, σ, ν) =µyKy+ν(α)

cy(ασ)y+νy!Kν( 1σ )

where α2 = 1σ2 + 2µ

cσ , and c = Rν(1/σ) = Kν+1(1/σ) [Kν(1/σ)]−1 for y = 0, 1, 2, ...,∞ whereµ > 0 , σ > 0 and −∞ < ν < ∞ and Kλ(t) = 1

2

∫∞0

xλ−1 exp{− 12 t(x + x−1)}dx is the

modified Bessel function of the third kind. Note that the above parameterization is different fromStein, Zucchini and Juritz (1988) who use the above probability function but treat µ/c, α and ν asthe parameters.

Value

Returns a gamlss.family object which can be used to fit a Sichel distribution in the gamlss()function.

Note

The mean of the above Sichel distribution is µ and the variance is µ2[

2σ(ν+1)c + 1

c2 − 1]

Author(s)

Rigby, R. A., Stasinopoulos D. M. and Akantziliotou C.

References


Rigby, R. A., Stasinopoulos D. M. and Akantziliotou, C. (2006) Modelling the parameters of afamily of mixed Poisson distribtions including the Sichel and Delaptorte. Submitted for publication.


Stein, G. Z., Zucchini, W. and Juritz, J. M. (1987). Parameter Estimation of the Sichel Distributionand its Multivariate Extension. Journal of American Statistical Association, 82, 938-944.

See Also

gamlss, gamlss.family, PIG , SI

Examples

SICHEL()# gives information about the default links for the Sichel distribution#plot the pdf using plotplot(function(y) dSICHEL(y, mu=10, sigma=1, nu=1), from=0, to=100, n=100+1, type="h") # pdf# plot the cdfplot(seq(from=0,to=100),pSICHEL(seq(from=0,to=100), mu=10, sigma=1, nu=1), type="h") # cdf# generate random sample


ST3 23

tN <- table(Ni <- rSICHEL(100, mu=5, sigma=1, nu=1))r <- barplot(tN, col='lightblue')# fit a model to the datagamlss(Ni~1,family=SICHEL, control=gamlss.control(n.cyc=50))

ST3 The skew t distribution, type 3

Description

The Skew t type 3 distribution Jones and Faddy (2003). The functions dST3, pST3, qST3 andrST3 define the density, distribution function, quantile function and random generation for theskew t distribution type 3.

Usage

ST3(mu.link = "identity", sigma.link = "log", nu.link = "identity", tau.link="log")dST3(y, mu = 0, sigma = 1, nu = 0, tau=1, log = FALSE)pST3(q, mu = 0, sigma = 1, nu = 0, tau=1, lower.tail = TRUE, log.p = FALSE)qST3(p, mu = 0, sigma = 1, nu = 0, tau=1, lower.tail = TRUE, log.p = FALSE)rST3(n, mu = 0, sigma = 1, nu = 0, tau=1)

Arguments



nu.link Defines the nu.link, with "identity" link as the default for the nu parameter.Other links are "1/mu2" and "log"

tau.link Defines the nu.link, with "log" link as the default for the nu parameter. Otherlinks are "inverse", "identity"


mu vector of mu parameter values


nu vector of nu parameter values

tau vector of tau parameter values






24 ST3

Details

The probability density function of the skew t distribution type 3, (ST3), is defined as

f(y|µ, σ, ν, τ) =1c

[1 +

z

(a + b + z2)1/2

]a+1/2 [1− z

(a + b + z2)1/2

]b+1/2

where c = 2a+b−1(a + b)1/2B(a, b), and B(a, b) = Γ(a)Γ(b)/Γ(a + b) and z = (y − µ)/σ andν = (a − b)/ [ab(a + b)]1/2 and τ = 2/(a + b) for −∞ < y < ∞, −∞ < µ < ∞, σ > 0,−∞ < ν > ∞ and τ > 0.

Value

ST3() returns a gamlss.family object which can be used to fit the skew t type 3 distributionin the gamlss() function. dST3() gives the density, pST3() gives the distribution function,qST3() gives the quantile function, and rST3() generates random deviates.

Note

The mean of the ex-Gaussian is µ + ν and the variance is σ2 + ν2.

Author(s)

Bob Rigby and Mikis Stasinopoulos

References

Jones, M.C. and Faddy, M. J. (2003) A skew extension of the t distribution, with applications.Journal of the Royal Statistical Society, Series B, 65, pp 159-174.



See Also

gamlss, gamlss.family, BCCG, GA, IG LNO

Examples

y<- rST3(100, mu=10, nu=1, sigma=3)hist(y)m1<-gamlss(y~1, family=ST3)plot(m1)curve(dST3(y=x, mu=300 ,sigma=35,nu=100), 100, 600,main = "The ex-GAUS density mu=300 ,sigma=35,nu=100")plot(function(x) pST3(x, mu=300,sigma=35,nu=100), 100, 600,main = "The ex-GAUS cdf mu=300, sigma=35, nu=100")


WEI3 25

WEI3 A specific parameterization of the Weibull distribution for fitting aGAMLSS

Description

The function WEI3 can be used to define the Weibull distribution, a two parameter distribution, fora gamlss.family object to be used in GAMLSS fitting using the function gamlss(). Thisis a parameterization of the Weibull distribution where µ is the mean of the distribution. [Notethat the GAMLSS functions WEI and WEI2 use different parameterizations for fitting the Weibulldistribution.] The functions dWEI3, pWEI3, qWEI3 and rWEI3 define the density, distributionfunction, quantile function and random generation for the specific parameterization of the Weibulldistribution.

Usage

WEI3(mu.link = "log", sigma.link = "log")dWEI3(y, mu = 1, sigma = 1, log = FALSE)pWEI3(q, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)qWEI3(p, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)rWEI3(n, mu = 1, sigma = 1)

Arguments


sigma.link Defines the sigma.link, with "log" link as the default for the sigma parame-ter, other link is the "inverse" and "identity"


mu vector of the mu parameter values

sigma vector of sigma parameter values





Details

The parameterization of the function WEI3 is given by

f(y|µ, σ) =σ

β

(y

β

)σ−1

e−( yβ )σ

where β = µΓ((1/σ)+1) for y > 0, µ > 0 and σ > 0. The GAMLSS functions dWEI3, pWEI3,

qWEI3, and rWEI3 can be used to provide the pdf, the cdf, the quantiles and random generated

26 WEI3

numbers for the Weibull distribution with argument mu, and sigma. [See the GAMLSS functionWEI for a different parameterization of the Weibull.]

Value

WEI3() returns a gamlss.family object which can be used to fit a Weibull distribution inthe gamlss() function. dWEI3() gives the density, pWEI3() gives the distribution function,qWEI3() gives the quantile function, and rWEI3() generates random deviates. The latest func-tions are based on the equivalent R functions for Weibull distribution.

Warning

In WEI3 the estimated parameters mu and sigma can be highly correlated so it is advisable to usethe CG() method for fitting [as the RS() method can be veru slow in this situation.]

Note

The mean in WEI3 is given by µ and the variance µ2{

Γ(2/σ + 1)/ [Γ(1/σ + 1)]2 − 1}

Author(s)

Bob Rigby 〈[email protected]〉 and Mikis Stasinopoulos 〈[email protected]〉

References



See Also

gamlss, gamlss.family, WEI, WEI2

Examples

WEI3()dat<-rWEI(100, mu=.1, sigma=2)gamlss(dat~1, family=WEI3, method=CG())


ZIP2 27

ZIP2 Zero inflated poisson distribution for fitting a GAMLSS model

Description

The function ZIP2 defines the zero inflated Poisson type 2 distribution, a two parameter distribu-tion, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss().The functions dZIP2, pZIP2, qZIP2 and rZIP2 define the density, distribution function, quan-tile function and random generation for the inflated poisson, ZIP2(), distribution. The ZIP2 is adifferent parameterization of the ZIP distribution. In the ZIP2 the mu is the mean of the distribution.

Usage

ZIP2(mu.link = "log", sigma.link = "logit")dZIP2(y, mu = 5, sigma = 0.1, log = FALSE)pZIP2(q, mu = 5, sigma = 0.1, lower.tail = TRUE, log.p = FALSE)qZIP2(p, mu = 5, sigma = 0.1, lower.tail = TRUE, log.p = FALSE)rZIP2(n, mu = 5, sigma = 0.1)

Arguments

mu.link defines the mu.link, with "log" link as the default for the mu parameter

sigma.link defines the sigma.link, with "logit" link as the default for the sigma param-eter which in this case is the probability at zero. Other links are "probit" and"cloglog"’(complementary log-log)

y vector of (non-negative integer) quantiles

mu vector of positive means

sigma vector of probabilities at zero


q vector of quantiles

n number of random values to return

log, log.p logical; if TRUE, probabilities p are given as log(p)


Details

Let Y = 0 with probability σ and Y ∼ Po(µ/ [1− σ]) with probability (1− σ) then Y has a Zeroinflated Poisson type 2 distribution given by

f(y|µ, σ) = σ + (1− σ)e−µ/(1−σ) if y = 0

f(y|µ, σ) = (1− σ)e−µ/(1−σ) [µ/(1− σ)]y

y!if y = 1, 2, 3, . . .

The mean of the distribution in this parameterization is mu.

28 exGAUS

Value

returns a gamlss.family object which can be used to fit a zero inflated poisson distribution inthe gamlss() function.

Note

Author(s)

Bob Rigby, Gillian Heller and Mikis Stasinopoulos

References

Lambert, D. (1992), Zero-inflated Poisson Regression with an application to defects in Manufactur-ing, Technometrics, 34, pp 1-14.



See Also

gamlss, gamlss.family, ZIP

Examples

ZIP2()# gives information about the default links for the normal distribution# creating data and plotting themdat<-rZIP2(1000, mu=5, sigma=.1)r <- barplot(table(dat), col='lightblue')# fit the disteibutionmod1<-gamlss(dat~1, family=ZIP2)# fits a constant for mu and sigmafitted(mod1)[1]fitted(mod1,"sigma")[1]

exGAUS The ex-Gaussian distribution

Description

The ex-Gaussian distribution is often used by psychologists to model response time (RT). It isdefined by adding two random variables, one from a normal distribution and the other from anexponential. The parameters mu and sigma are the mean and standard deviation from the normaldistribution variable while the parameter nu is the mean of the exponential variable. The functionsdexGAUS, pexGAUS, qexGAUS and rexGAUS define the density, distribution function, quantilefunction and random generation for the ex-Gaussian distribution.


exGAUS 29

Usage

exGAUS(mu.link = "identity", sigma.link = "log", nu.link = "log")dexGAUS(y, mu = 5, sigma = 1, nu = 1, log = FALSE)pexGAUS(q, mu = 5, sigma = 1, nu = 1, lower.tail = TRUE, log.p = FALSE)qexGAUS(p, mu = 5, sigma = 1, nu = 1, lower.tail = TRUE, log.p = FALSE,

lower.limit = mu - 10 * sqrt(sigma^2 + nu^2),upper.limit = mu + 10 * sqrt(sigma^2 + nu^2))

rexGAUS(n, mu = 5, sigma = 1, nu = 1, ...)

Arguments



nu.link Defines the nu.link, with "log" link as the default for the nu parameter. Otherlinks are "inverse", "identity" and "logshifted" (shifted from one)


mu vector of mu parameter values


nu vector of nu parameter values






lower.limit the q function uses a search for finding quantiles given the probabilities. lower.limitis the lowest limmit of the search

upper.limit The upper limit of the search

Details

The probability density function of the ex-Gaussian distribution, (exGAUS), is defined as

f(y|µ, σ, ν) =1ν

eµ−y

ν + σ2

2ν2 Φ(y − µ

σ− σ

ν)

where Φ is the cdf of the standard normal distribution, for −∞ < y < ∞, −∞ < µ < ∞, σ > 0and ν > 0.

Value

exGAUS() returns a gamlss.family object which can be used to fit ex-Gaussian distributionin the gamlss() function. dexGAUS() gives the density, pexGAUS() gives the distributionfunction, qexGAUS() gives the quantile function, and rexGAUS() generates random deviates.

30 gamlss.dist-package

Note

The mean of the ex-Gaussian is µ + ν and the variance is σ2 + ν2.

Author(s)

Mikis Stasinopoulos and Bob Rigby

References

Cousineau, D. Brown, S. and Heathecote A. (2004) Fitting distributions using maximum likelihood:Methods and packages, Behavior Research Methods, Instruments and Computers, 46, 742-756.



See Also

gamlss, gamlss.family, BCCG, GA, IG LNO

Examples

y<- rexGAUS(100, mu=300, nu=100, sigma=35)hist(y)m1<-gamlss(y~1, family=exGAUS)plot(m1)curve(dexGAUS(y=x, mu=300 ,sigma=35,nu=100), 100, 600,main = "The ex-GAUS density mu=300 ,sigma=35,nu=100")plot(function(x) pexGAUS(x, mu=300,sigma=35,nu=100), 100, 600,main = "The ex-GAUS cdf mu=300, sigma=35, nu=100")

gamlss.dist-packageThe GAMLSS distributions

Description

This package contains additional distributions to be used for GAMLSS models. Each distributionshas its probability function, d, its commutative probability function, p, the inverse of the commu-tative probability function, q, its random generation function, r, and also the gamlss.familygenerating function


parzen 31

Details

Package: gamlss.distType: PackageVersion: 1.5.0Date: 2006-12-13License: GPL (version 2 or later)

This package is to be used with the package gamlss.

Author(s)

Mikis Stasinopoulos <[email protected]>, Bob Rigby <[email protected]>with contributions from Calliope Akantziliotou and Raydonal Ospina <[email protected]>.

Maintainer: Mikis Stasinopoulos <[email protected]>

References


Stasinopoulos D. M., Rigby R.A. and Akantziliotou C. (2003) Instructions on how to use theGAMLSS package in R. Accompanying documentation in the current GAMLSS help files, (seealso http://www.gamlss.com/)

See Also


Examples

plot(function(y) dSICHEL(y, mu=10, sigma = 0.1 , nu=1 ), from=0, to=30, n=30+1, type="h") # pdfpdf.plot(family=SICHEL, mu=10, sigma=0.1, nu=1, min=0, max=30, step=1)# cdf plotPPP <- par(mfrow=c(2,1))plot(function(y) pSICHEL(y, mu=10, sigma =0.1, nu=1 ), from=0, to=30, n=30+1, type="h") # cdfcdf<-pSICHEL(0:30, mu=10, sigma=0.1, nu=1)sfun1 <- stepfun(1:30, cdf, f = 0)plot(sfun1, xlim=c(0,30), main="cdf(x)")par(PPP)

parzen Data files for GAMLSS


32 parzen

Description

These are several small data files usefull for gamlss fits.

Parzen: Parzen (1979) and also contained in Hand et al. (1994), data set 278. The data give theannual snowfall in Buffalo, NY (inches) for the 63 years, from 1910 to 1972 inclusive.

glass: show the strength of glass fibres, measured at the National Physical Laboratory, England,see Smith and Naylor (1987), (the unit of measurement were not given in the paper).

tensile: These data come from Quesenberry and Hales (1980) and were also reproduced in Handet al. (1994), data set 180, page 140. They contain measurements of tensile strength of polyesterfibres and the authors were trying to check if they were consistent with the lognormal distribution.According to Hand et al. (1994) "these data follow from a preliminary transformation. If thelognormal hypothesis is correct, these data should have been uniformly distributed".

margolin: Margolin et al. (1981) present data from an Ames Salmonella assay, where y is the num-ber of revertant colonies observed on a plate given a dose y of quinoline. The data were subsequentlyanalysed by Breslow (1984), Lawless (1987) and Saha and Paul (2005).

computing: The data relate to DEC-20 computers which operated at the Open University in the1980. They give the number of computers that broke down in each of the 128 consecutive weeks ofoperation, starting in late 1983, see Hand et al. (1994) page 109 data set 141.

lice : The data come from Williams (1944) and they are lice per head of Hindu male prisoners inCannamore, South India, 1937-1939.

alveolar : alveolar-bronchiolar adenomas data used by Tamura and Young (1987) and also reproducein Hand et al. (1994), data set 256. The data are the number of mice out of certain number of mice(the binomial denominator) in 23 independent groups, having alveolar-bronchiolar adenomas.

species: The number of different fish species (y=fish) was recorded for 70 lakes of the worldtogether with explanatory variable x=log(lake) area. The data are given and analyzed by Steinand Juritz (1988).

CD4: The data were given by Wade and Ades (1994) and refer to cd4 counts from uninfectedchildren born to HIV-1 mothers and the age of the child.

LGAclaims: the data were given by Gillian Heller and can be found in de Jong and Heller (2007).This data set records the number of third party claims, Claims, in a twelve month period between1984-1986 in each of 176 geographical areas (local government areas) in New South Wales, Aus-tralia. Areas are grouped into thirteen statistical divisions (SD). Other recorded variables are thenumber of accidents, Accidents, the number of people killed or injured and population with allvariables classified according to area.

Usage

data(parzen)data(glass)data(tensile)data(margolin)data(computer)data(lice)data(alveolar)data(species)data(CD4)

parzen 33

data(LGAclaims)

Format

Data frames each with the following variable.

snowfall the annual snowfall in Buffalo, NY (inches) for the 63 years, from 1910 to 1972 in-clusive, 63 observations

strength a numeric vector showing the strength of glass fibres

str a numeric vector showing the tensile strength

y a numeric vector showing the number of revertant colonies observed on a plate given a dose x ofquinoline.

x a numeric vector showing a a dose x of quinoline.

failure a numeric vector showing the number of times computers failed

head a numeric vector showing the number lice per head of Hindu male prisoners in Cannamore,South India, 1937-1939.

freq a numeric vector showing the frequency of lice per head

r a numeric vector showing the number of mice out of n number of mice (the binomial denominatorbelow) in 23 independent groups, having alveolar-bronchiolar adenomas.

n a numeric vector showing the total number of mice

fish a numeric vector showing the number of different species in 70 lakes in the word

lake a numeric vector showing the lake area

cd4 a numeric vector showing the CD4 counts

age the age of the child

Claims the number of third party claims

LGA Local government areas in New South Wales

SD statistical divisions

Pop_density population density

KI the number of people killed or injured

Accidents the number of accidents

Population population size

L_KI log of KI

L_Accidents the log of the number of accidents

L_Population log Population

Details

Data sets usefull for the GAMLSS booklet

34 parzen

References

Breslow, N. (1984) Extra-Poisson variation in log-linear models. Applied Statistics, 33, 38-44.

de Jong, P. and Heller G. (2007) Generalized Linear Models for Insurance Data , Cambridge Uni-versity Press

Hand et al. (1994) A handbook of small data sets. Chapman and Hall, London.

Lawless, J.F. (1987) Negative binomial and mixed Poisson regression. The Canadian Journal ofStatistics, 15, 209-225.

Margolin, B.H., Kaplan, N. and Zeiger, E. (1981) Statistical analysis of the Ames salmonella/microsometest. Proceedings of the National Academy of Science, U.S.A., 76, 3779-3783.

Quesenberry, C. and Hales, C. (1980). Concentration bands for uniformily plots. Journal of Statis-tical Computation and Simulation, 11, 41:53.

Parzen E. (1984) Nonparamemetric statistical daya modelling. JASA, 74, 105-131.

Saha, K. and Paul, S. (2005) Bias-Corrected Maximum Likelihood Estimator of the Negative Bino-mial Dispersion Parameter. Biometrics, 61, 179-185

Smith R. L. Naylor, J. C. (1987) A comparison of maximum likelihood and Bayesian estimators forthe three-parameter Weibull distributuion. Appl. Statist. 36, 358-369

Stein, G. Z. and Juritz, J. M. (1988). Linear models with an inverse Gaussian-Poisson error distri-bution. Communications in Statistics- Theory and Methods, 17, 557-571.

Wade, A. M. and Ader, A. E. (1994) Age-related reference ranges : Significance tests for modelsand confidence intervals for centiles. Statistics in Medicine, 13, pages 2359-2367.

Examples

data(parzen)with(parzen, hist(snowfall))data(glass)with(glass, hist(strength))data(tensile)with(tensile,hist(str))data(margolin)with(margolin, plot(y~x))data(computer)with(computer, plot(table(failure)))data(lice)with(lice, plot(freq~head, type="h"))data(alveolar)with(alveolar, hist(r/n))data(species)with(species, plot(fish~log(lake)))data(CD4)with(CD4,plot(cd4~age))data(LGAclaims)with(LGAclaims, plot(data.frame(Claims, Pop_density, KI, Accidents, Population)))

Index

∗Topic datasetsparzen, 31

∗Topic distributionBEOI, 1BEZI, 4DEL, 6exGAUS, 28EXP, 8gamlss.dist-package, 30GG, 10GIG, 12NOF, 14RGE, 16SHASH, 18SICHEL, 20ST3, 22WEI3, 24ZIP2, 26

∗Topic packagegamlss.dist-package, 30

∗Topic regressionBEOI, 1BEZI, 4DEL, 6exGAUS, 28EXP, 8GG, 10GIG, 12NOF, 14RGE, 16SHASH, 18SICHEL, 20ST3, 22WEI3, 24ZIP2, 26

alveolar (parzen), 31

BCCG, 24, 29BCT, 20

BEOI, 1, 3BEZI, 4, 6

CD4 (parzen), 31computer (parzen), 31

dBEOI (BEOI), 1dBEZI (BEZI), 4dDEL (DEL), 6DEL, 6dexGAUS (exGAUS), 28dEXP (EXP), 8dGG (GG), 10dGIG (GIG), 12dNOF (NOF), 14dRGE (RGE), 16dSHASH (SHASH), 18dSICHEL (SICHEL), 20dST3 (ST3), 22dWEI3 (WEI3), 24dZIP2 (ZIP2), 26

exGAUS, 28EXP, 8

GA, 11, 24, 29gamlss, 3, 6, 8, 9, 11, 13, 15, 18, 20, 22, 24,

26, 27, 29, 31gamlss.dist

(gamlss.dist-package), 30gamlss.dist-package, 30gamlss.family, 3, 6, 8, 9, 11, 13, 15, 18,

20, 22, 24–27, 29, 31GG, 10GI, 13GIG, 12glass (parzen), 31

IG, 24, 29

JSU, 20

35

36 INDEX

LGAclaims (parzen), 31lice (parzen), 31LNO, 24, 29

margolin (parzen), 31meanBEOI (BEOI), 1meanBEZI (BEZI), 4

NO, 15NOF, 14

parzen, 31pBEOI (BEOI), 1pBEZI (BEZI), 4pDEL (DEL), 6pexGAUS (exGAUS), 28pEXP (EXP), 8pGG (GG), 10pGIG (GIG), 12PIG, 22plotBEOI (BEOI), 1plotBEZI (BEZI), 4pNOF (NOF), 14pRGE (RGE), 16pSHASH (SHASH), 18pSICHEL (SICHEL), 20pST3 (ST3), 22pWEI3 (WEI3), 24pZIP2 (ZIP2), 26

qBEOI (BEOI), 1qBEZI (BEZI), 4qDEL (DEL), 6qexGAUS (exGAUS), 28qEXP (EXP), 8qGG (GG), 10qGIG (GIG), 12qNOF (NOF), 14qRGE (RGE), 16qSHASH (SHASH), 18qSICHEL (SICHEL), 20qST3 (ST3), 22qWEI3 (WEI3), 24qZIP2 (ZIP2), 26

rBEOI (BEOI), 1rBEZI (BEZI), 4rDEL (DEL), 6rexGAUS (exGAUS), 28

rEXP (EXP), 8RGE, 16rGG (GG), 10rGIG (GIG), 12rNOF (NOF), 14rRGE (RGE), 16rSHASH (SHASH), 18rSICHEL (SICHEL), 20rST3 (ST3), 22rWEI3 (WEI3), 24rZIP2 (ZIP2), 26

SHASH, 18SI, 8, 22SICHEL, 8, 20species (parzen), 31ST3, 22

tensile (parzen), 31

WEI, 24, 26WEI2, 24, 26WEI3, 24

ZIP, 27ZIP2, 26

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