Post on 14-Aug-2020
A goodness-of-fit test based on the multiplier Bootstrap with application to left-truncated data
by Takeshi Emura
Joint work with Dr. Yoshihiko Konno,Japan Women’s University
1
Seminar at Institute of StatisticsSeminar at Institute of StatisticsNational University of KaohsiungNational University of Kaohsiung
Outlines• Goodness-of-fit test: Review• Multiplier Bootstrap: Review
• Goodness-of-fit with application toleft-truncated data1. Proposed method2. Simulation 3. Data analysis
2
Goodness-of-fit: Review•Parametric family:
•Underlying CDF =
•Testing
based on data
3
pF Rθθ ,}|{
F
FHvsFH :.: 10
FXX n ~...,,1
•Under , there exists such that .
is consistent
Test Statistics1) Kolmogorov-Smirnov statistic
2) Cramér-von Mises statistic
4
|,)()(ˆ|sup ˆ xFxFKkRx
θ
)(})()(ˆ{ ˆ2
ˆ xdFxFxFC θθ
),...,(ˆˆ1 nXXθθ
FH :0
θ θFF
)()(1)(ˆ
),()()(
1
ˆ
xFxXIn
xF
xFxFxFn
ii
θθ
•The null distribution of :
depend on both and
1) Table available in Lilliefors (1967 JASA)
2) Table available in Lilliefores (1969 JASA)
5
|)()(ˆ|sup ˆ xFxFKkRx
θ
)(})()(ˆ{ ˆ2
ˆ xdFxFxFC θθ
θ }|{ θθF
}0,|),({ 22 RN
}0|)({ Exp
•Parametric Bootstrap has been suggested to approximate the null distribution of
Under cerntain regularity conditions, the parametric bootstrap is theoretically and numerically justified:• Stute et al. (1993 Metrika): Continuous case• Henze (1996 Can. J. Stat): Discrete case• Genest & Remillard (2008 Annales de Inst. Poincare) : Easy-to-check regularity conditions
• p.279 of Asymptotic Statistics by van der Vaart (1998): Empirical process approach
6
|)()(ˆ|sup ˆ xFxFKkRx
θ
)(})()(ˆ{ ˆ2
ˆ xdFxFxFC θθ
Algorithm:
7
0.05 )(IB1 value-
if }|{:HReject :4 Step
)(})()(ˆ{
Compute :3 Step),,( from
)(I1)(ˆ and ˆ Calculate :2 Step
,...,1,~),,( Generate :1 Step
B
1b
)*(
0
ˆ2
ˆ)*()*(
)*()*(1
n
1i
)*(1
)*()*(
ˆ)*()*(
1
)(*)(*
CCP
F
xdFxFxFC
XX
xXn
xF
BbFXX
b
bb
bn
b
bbb
bn
b
bb
θ
θ
θ
θθ
θ
•The parametric Bootstrap is very time consuming,especially when is multivariate
•A tool to reduce the computational cost “Multiplier Bootstrap” or “weighted Bootstrap”become very popular in the recent literature
(Remillard & Scaillet 2009 JMVA; Kojadinovic & Yan 2011 Stat.& Computing;Bucher & Dette 2010 Stat&Prob letters; Kojadinovic & Yan, 2012 manusc.)
•I use 4 slide to explain what is “multiplier Bootstrap”
8
F
What is multiplier Bootstrap?• IID: • Empirical Process:
How to approximate the distribution ?• Re-sampling:
Bootstrap version:
9
FXX n ~,...,1
i
in tFtXIn
t )}()({1)(G
),...,(),...,( **11 nn XXXX
i
in tFtXIn
t )}(ˆ)({1)( **G i
i tXIn
tF )(1)(ˆ
What is multiplier Bootstrap?• Multiplier representation
• Multiplier Bootstrap is a modification :
10
} )/1...,,/1(, { lMultinomia~),...,( where
)}(ˆ)({1
)}(ˆ)({1)(
1
**
nnnZZ
tFtXIZn
tFtXIn
t
n
iii
iin
G
1Var(Z) 0,E[Z] Z,~),...,( 1 iid
nZZ
i
iin tFtXIZn
t )}(ˆ)({1)(MPG
Ref: Sec 3.6 of van der Vaart & Wellner (1996) Weak convergence & empirical processes
----- Original----- Multiplier Boot
11
)(MP tnG)(tnG
0.0 0.5 1.0 1.5 2.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
F_n(
t)
n=1000, X~Exp(1), Z~N(0,1)
12
ondistributi asymptotic same thehas )( and )(: oremlimit the central Multiplier
MP tt nn FF
i
iUn
T 1 :Statistic
• In principle, Multiplier Bootstrap is applicable to any statistics of the sum of iid terms:
i
iiUZn
T 1 : versionBootstrap MP *
( Ref: Sec 2.9: Multiplier Central Limit Theorem in the book:van der Vaart & Wellner (1996) Weak convergence & empirical processes )
From now on, I focus on applications to left-truncated data
13
Left-Truncated data• Vast literature: see
Book of Klein & Moeschberger 2nd ed. 2003
14
jjjj XLnjXL subject to)},,1( ),({
0 2 4 6 8 10
02
46
810
Truncated Normal Plot
L
X L~N(5,2), X~N(5,2), Cov(L,X)=0.5
• Mathematical Set up Population random variable;
Post-truncated data;
15
),( OO XL
! observed nothing , If OO XL
observed is ),(),(pair a , If jjOOOO XLXLXL
jjjj XLnjXL subject to)},,1( ),({
)Pr()(),(
)|,Pr(
from ...
OOXLOOOO
XLxlxlf
XLdxXdlL
diiOO
I
• Likelihood
• Maximum likelihood estimator
Consistency & Asymptotic normality follows when andare 3 times continuously differentiable (Emura & Konno 2012 CSDA)
16
.),()Pr((),(for density :
where
,),(()(
xl
OO
OO
jjj
n
dldxxlfXLcXLf
XLfcL
θ
θ
θ
θ)
θ)θ
0θθ
θ )(log :ˆ L
θ)(c θf
Some useful bivariate modelsExample 1: Bivariate t or bivariate normal
Example 2: Bivariate Poisson (Holgate, 1964 Biometrika) :
Example 3: Zero-modified Poisson (Dietz and Böhning, 2000 CSDA)
17
2/)2(22/1222 ),|,(1)2/(
}2/)2{()(),(
v
LXXL
vxlQ
vvvxlf Σμ
θ
),,,),,(( 22 vLXXLXL μθ
tvvXLcLXLX
LXOO -student of cdf :)(: ,:2
)Pr()(22
θ
l
w
wwxX
wlL
wwxwlexlf XL
0 !)!()!(),(
θ
0
)(!
)Pr()(l
V
lLOO lS
leXLc XL
θ
lv
vX
V velS
X
!)(
,2,1,0;1,0,!
)()1(),( 1
xlx
eppxlfXx
Xll
θ
XpeXLc OO 1)Pr()(θ
18
Let }|{ θθf be a given parametric family.
fH :0 against fH :1 .
Goodness-of-fit• Cramér-von Mises statistic
* I will not use Kolmogorov-Smirnov statistic since it is computationally more demanding.
19
j jj
jjjjj
xl
nxXlLxlF
cXLFXLF
xlFdcxlFxlFC
/),(),(ˆ where
,})ˆ(/),(),(ˆ{
),(ˆ})ˆ(/),(),(ˆ{
2ˆ
2ˆ
I
θ
θ
θ
θ
)Pr(( OO XLc θ)
.discrete ),(
;continuous),(),( ,
lu xvu
xvulu
vuf
dudvvufxlF
θ
θ
θ
OL
OX
Some useful bivariate modelsExample 1: Bivariate normal
Example 2: Bivariate Poisson (Holgate, 1964) :
Example 3: Zero-modified Poisson (Dietz and Böhning, 2000)
20
density normal Bivariate),( xlfθ
l
w
wwxX
wlL
wwxwlexlf XL
0 !)!()!(),(
θ
,2,1,0;1,0,!
)()1(),( 1
xlx
eppxlfXx
Xll
θ
dsssssxxlFLLl
LLXX
LLXXLL
LLXX
LLXX )(/
//
/),(/)(
222222
θ
l
u
x
uv
u
w
wwvX
wuL
wwvwuexlF XL
0 0 !)!()!(),(
θ
,1 if!/
;0 if )!/()1(),(
0
0
lpue
luepxlF x
u
xX
x
u
xX
X
X
θ
Parametric Bootstrap
21
Step 0: Calculate the statistic i
iiii cXLFXLFC 2ˆ })ˆ(/),(),(ˆ{ θθ
.
Step 1: Generate ),( )()( bj
bj XL which follows the truncated distribution of ),(ˆ xlFθ ,
subject to )()( bj
bj XL , for njBb ,,2,1,,,2,1 .
Step 2: Calculate },,2,1;{ )*( BbC b ,
where i
bbi
bi
bi
bi
bb cXLFXLFC b2)()()(
ˆ)()()()*( })ˆ(/),(),(ˆ{ )( θθ and where
),(ˆ )( xlF b and )(ˆ bθ are the empirical CDF and MLE based on
},,2,1);,({ )()( njXL bj
bj .
Step 3: Reject 0H at the 100 % significance level if
BCCB
b
b /)(1
)*(I .
Parametric Bootstrap
22
Table 1: Type I error rates of the the Cramér-von-Mises type goodness-of-fit test
at level based on 300 replications. The cut-off value is obtained by the
parametric bootstrap-based procedure based on 1,000 resamplings.
OOYX
-0.70 -0.35 0.00 0.35 0.70
10.0 0.113 0.090 0.107 0.123 0.083
05.0 0.056 0.040 0.047 0.060 0.050
01.0 0.013 0.007 0.007 0.023 0.013
2
2
2
2
16.8116.8119.6416.8119.6419.64
60.8262.63-120
~LX
LX
XLX
LXL
X
LO
O
NXL
•Some simulation results under bivariate normal model (Emura & Konno, 2012a Stat Papers)
•Parametric Bootstrap is computationally very intensive, especially in Step 2:
• Maximizing likelihood functions B times oftenencounter the problem of local maxima(especially serious when data are not from the null)
23
Step 2: Calculate },,2,1;{ )*( BbC b ,
where i
bbi
bi
bi
bi
bb cXLFXLFC b2)()()(
ˆ)()()()*( })ˆ(/),(),(ˆ{ )( θθ
and where
),(ˆ )( xlF b and )(ˆ bθ are the empirical CDF and MLE based on
},,2,1);,({ )()( njXL bj
bj .
Alternatively, we apply the “Multiplier Bootstrap” to approximate the distribution of
Motivation: Reduce the computational cost
24
jjjjj
xl
cXLFXLF
xlFdcxlFxlFC
2ˆ
2ˆ
})ˆ(/),(),(ˆ{
),(ˆ})ˆ(/),(),(ˆ{
θ
θ
θ
θ
Theorem 1 (Emura & Konno 2012 CSDA):
Conditional on data, only random quantities are
25
θθθθθθθg
θθgθIθ
θθ
θθ
θθθ
θθ
θ
/)()(,/),(),(
},)(),()(),({)(),(
),(}/)(){,()(/),(),();,(ˆ where
),1()ˆ;,(ˆ1})ˆ(/),(),(ˆ{
2
1
ˆ
ccxlFxlFcxlFcxlFcxl
lnixlcxlFxXlLxlV
oxlVn
cxlFxlFn
jnjjj
Pj
j
1)( ,0)(E with variablerandom be Let jjj ZVarZZ
},,2,1;{ njZ j
j
jj xlVZn
)ˆ;,(ˆ1 θ
26
Step 0: Calculate the statistic i
iiii cXLFXLFC 2ˆ })ˆ(/),(),(ˆ{ θθ
and matrix
)ˆ(ˆ θV .
Step 1: Generate njBbNZ bj ,,2,1,,,2,1);1,0(~)( .
Step 2: Calculate },,2,1;{ )( BbC b ,
where ||/)ˆ(ˆ)(|| )()( nC bb θVZ and ),,( )()(1
)( bn
bb ZZ Z .
Step 3: Reject 0H at the 100 % significance level if
BCCB
b
b /)(1
)(I .
22
2
2ˆ
)ˆ(ˆ)ˆ;,(ˆ1
by edapproximat is })ˆ(/),(),(ˆ{
ofon dsitributi therem,limit theo central multiplier By the
nXLVZ
n
cXLFXLFC
i jiijj
jjjjj
θVZθ
θθ
27
Let })ˆ(/),(),(ˆ{),( ˆ θθ cxlFxlFnxlGn and j j
bj
bn xlVZnxlG )ˆ;,(ˆ),( )(2/1)( θ
for Bb ,,2,1 .
Theorem 2: Suppose that (R1) through (R8) listed in Emura & Konno (2012 CSDA)
hold. Under 0H , and given B,
)...,,,()...,,,( )()1()()1( BBnnn GGGGGG θθθ
in )1(2}),{( BD , where θG is the mean zero Gaussian process whose
covariance for 2** ),(),(),,( xlxl is given as
),,()(),()(/),(),()(/),(
});,();,({}),(),,({**12****
****
xlIxlcxlFxlFcxxllF
xlVxlVExlGxlGCov jj
θθθθθ
θθ
gθgθθ
θθ
where ),min( baba , and )()1( ...,, BGG θθ are independent copies of θG .
Simulation under the null
28
60.8262.63-120
X
L
μ ,
2
2
2
2
16.8116.8119.6416.8119.6419.64
LX
LX
XLX
LXL
Σ
• Model for : Bivariate normal
• Generate data:
• Compute Cramér-von Mises statistic
• Compute Bootstrap version
),( OO XL
jjjj XLjXL subject to)}400,,1( ),({
j
jjjj cXLFXLFC 2ˆ })ˆ(/),(),(ˆ{ θθ
}1000,,2,1;*{ :Boot Parametric Usual}1000,,2,1;{ :Boot Multiplier
)(
)(
bCbC
b
b
29
Multiplier Boot vs. naïve Bootstrap with 400n
Elapsed time
(in second)
Resampling mean and
SD (in parenthesis)
Multiplier Parametric
Bootstrap Multiplier
Parametric
Bootstrap
OOYX =0.70 23.17 1402.31 0.0621 (0.0314) 0.0603 (0.0304)
OOYX =0.35 22.84 1463.01 0.0606 (0.0269) 0.0581 (0.0252)
OOYX =0.00 23.04 1539.29 0.0597 (0.0219) 0.0586 (0.0225)
OOYX =-0.35 23.37 1380.09 0.0598 (0.0234) 0.0577 (0.0209)
OOYX =-0.70 26.77 1296.32 0.0591 (0.0225) 0.0569 (0.0217)
Almost same null distribution
Power study
30
we generated data from the bivariate t-distribution (Lang et al., 1989) while we performed the goodness-of-fit test under the null hypothesis of the bivariate normal distribution.
Data analysisLaw school data in US(Efron & Tibshirani 1993, An Introduction to the Bootstrap)
• Average LSAT (the national law test)• Average GPA (graduate point average)
for N=82 American law schools
31
)GPA,(LSAT jj for Nj ...,,2,1
32
}82,...,1);LSAT,{GPA jjj
260 280 300 320 340
500
550
600
650
700
100*GPA
LSA
T
• We artificially define the truncation
33
! observed nothing ,900GPA100LSAT If (ii)sample theobserve we,900GPA100LSAT If (i)
Observed
Un observed
260 280 300 320 340
500
550
600
650
700
100*GPA
LSA
T
• We rewrite the truncation criterion
*
34
Observe }49,,2,1);,({ jXL jj ,
subject to jj XL , where jjL GPA100900 and jjX LSAT .
GPA100 - 900
LSAT where
GPA100 - 900 LSAT ,900 GPA 100LSAT
O
O
OO
LX
L X
35
560 580 600 620 640
500
550
600
650
700
900 - GPA
LSA
T
jj
jj
XLjXL
subject to}49,,2,1);,({
Un observed
bivariate normal fit ?
Computational time for the multiplier method = 1.25 secondComputational time for the parametric Bootstrap = 222.46 second
36
Multiplier
log(Cramer-von Mises statistic)
Freq
uenc
y
-4 -3 -2 -1 0 1
010
2030
40
p-value = 0.884
Parametric Bootstrap
log(Cramer-von Mises statistic)
Freq
uenc
y
-4 -3 -2 -1 0 1
020
4060
p-value = 0.645
Summary: Why multiplier reduce computational cost ?• The multiplier involves only arithmetic operations (multiply,
sum). On the other hand, the parametric Bootstrap involves nonlinear maximization to get
• Re-sampling from a truncated parametric model is involved:
On the other hand, the multiplier only requires generatingi.i.d. sequences
37
)(ˆ bθ
Accept-reject algorithm
(i) data ),( XL from the distribution function ),(ˆ xlFθ is generated;
(ii) if XL , we accept the sample and set ),(),( )()( XLXL bj
bj ;
otherwise we reject ),( XL and return to (i).
Thank you for your kind Thank you for your kind attentionattention• Bücher, A. and Dette, H. , 2010. A note on bootstrap approximations for the empirical copula process. Statist.
Probab. Lett. 80, 1925-1932.• Dietz, E., Böhning, D., 2000. On estimation of the Poisson parameter in zero-modified Poisson models.
Computational Statistics ＆ Data Analysis 34, 441-459.• Efron, B., Tibshirani, R. J., 1993. An Introduction to the Bootstrap. Chapman & Hall, London.• Emura, T., Konno, Y., 2012. A goodness-of-fit test for parametric models based on dependently truncated data,
Computational Statistics & Data Analysis 56, 2237-2250.• Emura, T., Konno, Y, 2012. Multivariate normal distribution approaches for dependently truncated data. Statistical
Papers 53 (No.1), 133-149.• Genest C., Remillard B., 2008. Validity of the parametric bootstrap for goodness-of-fit testing in semiparametric
models, Annales de Institut Henri Poicare –Probabilites et Statistiques 44, 1096-1127.• Henze N., 1996. Empirical-distribution-function goodness-of-fit tests for discrete models. Cand. J. Statist. 24, 81-93.• Lilliefors HW, 1967. On the Kolmogorov-Smirnov test for normality with mean and variance unknown. JASA 62
399-402.• Lilliefors HW, 1969. On the Kolmogorov-Smirnov test with exponential distribution with mean unknown. JASA 64
387-389.• Fang, K.-T., Kotz, S., Ng, K. W., 1990. Symmetric Multivariate and Related Distributions. Chapman & Hall, New York.• Holgate, 1964. Estimation of the bivariate Poisson distribution. Biometrika 51, 241-245.• Klein, J. P., Moeschberger, M. L., 2003. Survival Analysis: Techniques for Censored and Truncated Data. New York,
Springer.• Kojadinovic, I., Yan, J., Holmes, M., 2011. Fast large-sample goodness-of-fit tests for copulas. Statistica Sinica 21,
841-871.• Kojadinovic, I., Yan, J., 2011. A goodness-of-fit test for multivariate multiparameter copulas based on multiplier
central limit theorems. Statistics and Computing 21, 17-30.• Lang, K. L., Little, J. A., Taylor, J. M. G., 1989. Robust statistical modeling using the t distribution. Journal of the
American Statistical Association 84, 881-896.• Stute W et al. , 1993. Bootstrap based goodness-of-fit test. Metrika 40, 243-256.• van der Vaart. A. W., Wellner, J. A., 1996. Weak Convergence and Empirical Process, Springer Series in Statistics,
Springer-Verlag, New York.• J. P. Klein, M. L. Moeschberger, Survival Analysis: Techniques for Censored and Truncated Data, 2nd edition,
Springer, New York, 2003.38