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Journal ofStatistical Computation and Simulation . Vol. 81, No.7, July 2011,843-855
Taylor&FrancisGroup
Smooth tests for the gamma distribution B. De Boeck?", O. Thas a , J.C.W. Rayner" and D.l Bestb a Department ofApplied Mathematics, Biometrics and Process Control, Ghent University, B-9000 Gent, Belgium; b School ofMathematical and Physical Sciences, University ofNewcastle, NSW 2308, Australia
(Received 3 April 2009;final version received 30 November 2009)
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The gamma distribution is often used to model data with right skewness. Smooth tests of goodness of fit are proposed for this distribution. Their powers are compared with powers of the Anderson-Darling test and tests based on the empirical Laplace transform, the empirical moment generating function and the independence of the mean and coefficient of variation that characterizes the gamma distribution. r O
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Keywords: Anderson-Darling test; empirical Laplace transform; empirical moment generating function; generalized smooth tests; goodness offit .
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AMS Subject Classification: 62-02; 62F03
1.
Introduction
The two-parameter gamma distribution b > 0 has probability density function
rea, b) with shape parameter a >
f(x; a, b) = b-ar(a)-lx a- 1 exp(-xjb)
0 and scale parameter
for x c- 0,
(1)
and zero otherwise, where I't.) ;s the gamma function. It is often used to model positive contin uous data with right skewness. We consider the situation where a and b are unknown nuisance parameters. Lockhart and Stephens [1] and Stephens [2] proposed tests of goodness of fit for this distribution based on the empirical distribution function (EDF). These EDF tests assess the probability integral transformed observations for uniformity after replacing the nuisance param eters by their maximum likelihood estimators (MLE). In [3,4] empirical powers from simulation studies showed unequivocally that the Anderson-Darling test was the best performing EDF test. Recently, Henze and Meintanis [3] suggested tests based on the empirical Laplace transform (ELT) and Henze and Meintanis Kallioras et al. [4] presented tests offit based on the empirical moment generating function (EMGF). In these two papers the proposed tests were empirically compared with other goodness-of-fit tests for the gamma distribution in a simulation study. Henze and Meintanis [3] conclude that the ELT tests are almost always more powerful than the EDF tests. For the EMGF test the situation is less clear, but Kallioras et al. [4] claim it to be competitive with *Corresponding author. Email:
[email protected]
ISSN 0094-9655 printjISSN 1563-5163 online
© 2011 Taylor & Francis
DOl: 10.1080/00949650903520936
http://www.infonnaworld.com
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test statistic of ELTl). In particular, the tests based on ,~~ and S~ seem to outperform all other tests for small sample sizes. Also the AD test (A 2 ) has good overall performance with a slight advantage over the EMGF test (Q). The CHAR test (2) has overall poor power. Finally, the GS test (S4) has very diversified power properties. Sometimes it is the best performing test, but often it is the worst. In general, its performance gets worse when the estimated shape parameter decreases.
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B. De Boeck et al.
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Table 5. Simulated powers against Weibull alternatives for the smooth tests, and the GS, AD, ELTl, EMGF and CHAR tests. All powers are calculated using the bootstrap null distributions at the 5% nominal significance level. Smooth test
V'2
V'3
V'4
S'3
S'4
20 50 100
0.1198 0.2520 0.4182
0.1375 0.2612 0.4298
0.0948 0.2226 0.2382
0.1659 0.3474 0.5577
0.1632 0.3440 0.5463
20 50 100
0.0634 0.1094 0.1889
0.0680 0.0615 0.0616
0.0439 0.0412 0.0447
0.0589 0.0764 0.1245
0.0558 0.0644 0.1054
20 50 100
0.1541 0.3777 0.6740
0.1178 0.1697 0.2646
0.0781 0.1319 0.1667
0.1469 0.3116 0.5528
0.1254 0.2770 0.5163
n
GS
AD
ELTI
EMGF
CHAR
S4
A2
TO)
Q
Z
0.0169 0.0096 0.0133
0.1206 0.2737 0.4948
0.1285 0.2685 0.4758
0.0394 0.1590 0.2742
0.1056 0.1212 0.1461
0.0874 0.1110 0.1526
0.0652 0.0858 0.1395
0.0694 0.1139 0.1959
0.0508 0.0502 0.0662
0.0493 0.0647 0.1057
0.2132 0.4076 0.6581
0.1257 0.2676 0.5007
0.1147 0.3010 0.6118
0.1696 0.2921 0.4742
0.1110 0.2794 0.5510
W(0.5, I)
W(1.5,1)
W(3. I) .-< .-
