Kybernetika 47 no. 6, 814-839, 2011

Randomized goodness of fit tests

Friedrich Liese and Bing Liu

Abstract:

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.

Keywords:

goodness of fit tests with estimated parameters, Kolmogorov-Smirnov test, Cramér-von Mises test, randomization

Classification:

64E17, 62E20

paper.pdf

References:

  1. P. Billingsley: Covergence of Probability Measures. Wiley, New York 1986.   CrossRef
  2. S. Csörgő and J. J. Faraway: The exact and asymptotic distributions of Cramér-von Mises statistics. J. Roy. Statist. Soc. Ser. B (Methodological) 58 (1996), 1, 221-234.   CrossRef
  3. R. B. D'Agostino and M. A. Stephens: Goodness-of-fit Techniques. Marcel Decker, New York and Basel 1986.   CrossRef
  4. J. Durbin: Distribution theory for tests based on the sample distribution function. Regional Conference Series in Applied Mathematics 9, SIAM, Philadelphia 1973.   CrossRef
  5. J. Durbin: Weak convergence of the sample distribution function when parameters are estimated. Ann. Statist. 1 (1973), 279-290.   CrossRef
  6. M. Genz and E. Haeusler: Empirical processes with estimated parameters under auxiliary information. J. Comput. Appl. Math. 186 (2006), 191-216.   CrossRef
  7. J. Haywood and E. Khmaladze: On distribution-free goodness-of-fit tetsting of exponentiality. Econometrics 143 (2008), 5-18.   CrossRef
  8. A. Janssen: Asymptotic relative efficiency of tests at the boundary of regular statistical modells. J. Statist. Plann. Inference 126 (2004), 461-477.   CrossRef
  9. E. Khmaladze: Martingale approach in the theory of goodness of fit tests. Theory Probab. Appl. 24 (1981), 2, 283-302.   CrossRef
  10. F. Liese and K. J. Miescke: Statistical Decision Theory, Estimation, Testing and Selection. Springer-Verlag, New York 2008.   CrossRef
  11. M. Matsumoto and T. Nishimura: Mersenne-Twister: A 623-dimensionally equidistributed uniform pseudo-random generator. ACM Trans. Model. Comp. Simul. 8 (1998), 19, 3-30.   CrossRef
  12. D. Pollard: Convergence of Stochastic Processes. Springer-Verlag, New York 1984.   CrossRef
  13. S. S. Shapiro and R. S. Francia: Approximate analysis of variance test for normality. J. Amer. Statist. Assoc. 67 (1972), 215-216.   CrossRef
  14. G. Shorack and J. A. Wellner: Empirical Processes with Applications to Statistics. Wiley, New York 1986.   CrossRef
  15. M. A. Stephens: EDF statistics for goodness of fit and some comparisons. J. Amer. Statist. Assoc. 69 (1974), 730-737.   CrossRef
  16. M. A. Stephens: Goodness of fit for the extreme value distribution. Biometrika 64 (1976), 583-588.   CrossRef
  17. W. Stute, W. G. Manteiga and M. P. Quindimil: Bootstrap based goodness-of-fit-tests. Metrika 40 (1993), 243-256.   CrossRef
  18. A. W. van der Vaart and J. A. Wellner: Weak convergence and Empirical Processes. With Applications to Statstics. Second edition 2000. Springer, New York 1996.   CrossRef
  19. A. W. van der Vaart: Asymptotic Statistics. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge Univ. Press. 1998.   CrossRef