Abstract:
We consider goodness-of-fit tests for hypotheses about the forms of distributions and their membership in prescribed families of distributions. We first describe the classical tests based on empirical processes such as the omega-square tests of Cramér–von Mises–Smirnov and the Kolmogorov–Smirnov tests. We also consider Shapiro–Wilk tests. We devote a considerable amount of attention to testing the hypothesis that a random variable or vector is normal. We describe tests based on transformations of the empirical process, minimal distance tests and estimates, tests for symmetry, uniformity, and independence, and tests based on spacings. At the end we study methods of computing and the distribution functions of quadratic forms of normal random variables connected with tests of omega-square type. Bibliography: 372 titles.
Citation:
G. V. Martynov, “Statistical tests based on empirical processes and related questions”, Itogi Nauki i Tekhniki. Ser. Teor. Veroyatn. Mat. Stat. Teor. Kibern., 30, VINITI, Moscow, 1992, 3–112; J. Soviet Math., 61:4 (1992), 2195–2271
\Bibitem{Mar92}
\by G.~V.~Martynov
\paper Statistical tests based on empirical processes and related questions
\serial Itogi Nauki i Tekhniki. Ser. Teor. Veroyatn. Mat. Stat. Teor. Kibern.
\yr 1992
\vol 30
\pages 3--112
\publ VINITI
\publaddr Moscow
\mathnet{http://mi.mathnet.ru/intv80}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=1172613}
\zmath{https://zbmath.org/?q=an:0790.62041}
\transl
\jour J. Soviet Math.
\yr 1992
\vol 61
\issue 4
\pages 2195--2271
\crossref{https://doi.org/10.1007/BF01104102}
Linking options:
https://www.mathnet.ru/eng/intv80
https://www.mathnet.ru/eng/intv/v30/p3
This publication is cited in the following 18 articles:
G. V. Martynov, “Anderson–Darling statistic and its “inverse””, J. Commun. Technol. Electron., 61:6 (2016), 709
Gennady Martynov, Mathematical Statistics and Limit Theorems, 2015, 209
Ilia Negri, Li Zhou, “On goodness-of-fit testing for ergodic diffusion process with shift parameter”, Stat Inference Stoch Process, 17:1 (2014), 51
Anis Gassem, “On the goodness-of-fit testing for a switching diffusion process”, Comptes Rendus. Mathématique, 349:15-16 (2011), 897
Anirban DasGupta, Springer Texts in Statistics, Probability for Statistics and Machine Learning, 2011, 221
Anis Gassem, “On Cramér–von Mises type test based on local time of switching diffusion process”, Journal of Statistical Planning and Inference, 141:4 (2011), 1355
G. V. Martynov, “Multivariable goodness tests and approximation of the residues of quadratic forms”, Autom. Remote Control, 71:7 (2010), 1346–1357
A. Gassem, “Goodness-of-fit test for switching diffusion”, Stat Inference Stoch Process, 13:2 (2010), 97
Paul Deheuvels, Guennadi V. Martynov, “A Karhunen–Loeve decomposition of a Gaussian process generated by independent pairs of exponential random variables”, Journal of Functional Analysis, 255:9 (2008), 2363
R. Tahmasbi, S. Rezaei, “Change Point Detection in GARCH Models for Voice Activity Detection”, IEEE Trans. Audio Speech Lang. Process., 16:5 (2008), 1038
Anirban DasGupta, Springer Texts in Statistics, Asymptotic Theory of Statistics and Probability, 2008, 421
G. Martynov, M. Mesbah, “Goodness of Fit Test and Latent Distribution Estimation in the Mixed Rasch Model”, Communications in Statistics - Theory and Methods, 35:5 (2006), 921
Paul Deheuvels, Guennady Martynov, High Dimensional Probability III, 2003, 57
Paul Deheuvels, Goodness-of-Fit Tests and Model Validity, 2002, 463
Lajos Horváth, Agnieszka Jach, Piotr Kokoszka, Statistical Data Analysis Based on the L1-Norm and Related Methods, 2002, 229
István Berkes, Lajos Horváth, Empirical Process Techniques for Dependent Data, 2002, 195
Lajos Horváth, Gilles Teyssière, “Empirical process of the squared residuals of an arch sequence”, Ann. Statist., 29:2 (2001)
Paul Deheuvels, Guennady V. Martynov, “Cramer-von mises-type tests with applications to tests of independence for multivariate extreme-value distributions”, Communications in Statistics - Theory and Methods, 25:4 (1996), 871
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