Geodesic
Randomized goodness of fit tests
Kybernetika, Tome 47 (2011) no. 6, pp. 814-839 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

Classical goodness of fit tests are no longer asymptotically distributional free if parameters are estimated. For a parametric model and the maximum likelihood estimator the empirical processes with estimated parameters is asymptotically transformed into a time transformed Brownian bridge by adding an independent Gaussian process that is suitably constructed. This randomization makes the classical tests distributional free. The power under local alternatives is investigated. Computer simulations compare the randomized Cramér-von Mises test with tests specially designed for location-scale families, such as the Shapiro-Wilk and the Shenton-Bowman test for normality and with the Epps-Pulley test for exponentiality.
Classical goodness of fit tests are no longer asymptotically distributional free if parameters are estimated. For a parametric model and the maximum likelihood estimator the empirical processes with estimated parameters is asymptotically transformed into a time transformed Brownian bridge by adding an independent Gaussian process that is suitably constructed. This randomization makes the classical tests distributional free. The power under local alternatives is investigated. Computer simulations compare the randomized Cramér-von Mises test with tests specially designed for location-scale families, such as the Shapiro-Wilk and the Shenton-Bowman test for normality and with the Epps-Pulley test for exponentiality.
Classification : 62E20, 64E17
Keywords: goodness of fit tests with estimated parameters; Kolmogorov–Smirnov test; Cramér–von Mises test; randomization 
@article{KYB_2011_47_6_a1,
     author = {Liese, Friedrich and Liu, Bing},
     title = {Randomized goodness of fit tests},
     journal = {Kybernetika},
     pages = {814--839},
     year = {2011},
     volume = {47},
     number = {6},
     mrnumber = {2907844},
     zbl = {06047588},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/KYB_2011_47_6_a1/}
}
TY  - JOUR
AU  - Liese, Friedrich
AU  - Liu, Bing
TI  - Randomized goodness of fit tests
JO  - Kybernetika
PY  - 2011
SP  - 814
EP  - 839
VL  - 47
IS  - 6
UR  - http://geodesic.mathdoc.fr/item/KYB_2011_47_6_a1/
LA  - en
ID  - KYB_2011_47_6_a1
ER  - 
%0 Journal Article
%A Liese, Friedrich
%A Liu, Bing
%T Randomized goodness of fit tests
%J Kybernetika
%D 2011
%P 814-839
%V 47
%N 6
%U http://geodesic.mathdoc.fr/item/KYB_2011_47_6_a1/
%G en
%F KYB_2011_47_6_a1
Liese, Friedrich; Liu, Bing. Randomized goodness of fit tests. Kybernetika, Tome 47 (2011) no. 6, pp. 814-839. http://geodesic.mathdoc.fr/item/KYB_2011_47_6_a1/

[1] Billingsley, P.: Covergence of Probability Measures. Wiley, New York 1986. | MR

[2] Bowman, K. O., Shenton, L. R.: Omnibus test contours for departures from normality based on $\sqrt{b_{1}}$ and $b_{2}$. Biometrica 62 (1975), 243-250. | MR | Zbl

[3] Csörgő, S., Faraway, J. J.: The exact and asymptotic distributions of Cramér–von Mises statistics. J. Roy. Statist. Soc. Ser. B (Methodological) 58 (1996), 1, 221–234. | MR

[4] D’Agostino, R. B., Stephens, M. A.: Goodness-of-fit Techniques. Marcel Decker, New York and Basel 1986. | MR

[5] Durbin, J.: Distribution theory for tests based on the sample distribution function. Regional Conference Series in Applied Mathematics 9, SIAM, Philadelphia 1973. | MR | Zbl

[6] Durbin, J.: Weak convergence of the sample distribution function when parameters are estimated. Ann. Statist. 1 (1973), 279–290. | DOI | MR | Zbl

[7] Genz, M., Haeusler, E.: Empirical processes with estimated parameters under auxiliary information. J. Comput. Appl. Math. 186 (2006), 191–216. | DOI | MR | Zbl

[8] Haywood, J., Khmaladze, E.: On distribution-free goodness-of-fit tetsting of exponentiality. Econometrics 143 (2008), 5–18. | DOI | MR

[9] Janssen, A.: Asymptotic relative efficiency of tests at the boundary of regular statistical modells. J. Statist. Plann. Inference 126 (2004), 461–477. | DOI | MR

[10] Khmaladze, E.: Martingale approach in the theory of goodness of fit tests. Theory Probab. Appl. 24 (1981), 2, 283–302. | Zbl

[11] Liese, F., Miescke, K. J.: Statistical Decision Theory, Estimation, Testing and Selection. Springer-Verlag, New York 2008. | MR | Zbl

[12] Matsumoto, M., Nishimura, T.: Mersenne-Twister: A 623-dimensionally equidistributed uniform pseudo-random generator. ACM Trans. Model. Comp. Simul. 8 (1998), 19, 3–30. | DOI

[13] Pollard, D.: Convergence of Stochastic Processes. Springer-Verlag, New York 1984. | MR | Zbl

[14] Shapiro, S. S., Francia, R. S.: Approximate analysis of variance test for normality. J. Amer. Statist. Assoc. 67 (1972), 215–216. | DOI

[15] Shorack, G., Wellner, J. A.: Empirical Processes with Applications to Statistics. Wiley, New York 1986. | MR | Zbl

[16] Stephens, M. A.: EDF statistics for goodness of fit and some comparisons. J. Amer. Statist. Assoc. 69 (1974), 730–737. | DOI

[17] Stephens, M. A.: Goodness of fit for the extreme value distribution. Biometrika 64 (1976), 583–588. | DOI | MR

[18] Stute, W., Manteiga, W. G., Quindimil, M. P.: Bootstrap based goodness-of-fit-tests. Metrika 40 (1993), 243–256. | DOI | MR | Zbl

[19] Vaart, A. W. van der, Wellner, J. A.: Weak convergence and Empirical Processes. With Applications to Statstics. Second edition 2000. Springer, New York 1996. | MR

[20] Vaart, A. W. van der: Asymptotic Statistics. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge Univ. Press. 1998. | MR