×

zbMATH — the first resource for mathematics

Modeling statistic distributions for nonparametric goodness-of-fit criteria for testing complex hypotheses with respect to the inverse Gaussian law. (English. Russian original) Zbl 1203.62080
Autom. Remote Control 71, No. 7, 1358-1373 (2010); translation from Avtom. Telemekh. 2010, No. 7, 83-102 (2010).
Summary: We give percentage point tables and statistic distribution models for nonparametric goodness-of-fit criteria for testing complex hypothesis with respect to the inverse Gaussian law in case of using maximal likelihood estimates.

MSC:
62G10 Nonparametric hypothesis testing
62Q05 Statistical tables
62E15 Exact distribution theory in statistics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Lemeshko, B.Yu. and Postovalov, S.N., On the Distributions of Nonparametric Goodness-of-Fit Criteria for Estimating on the Samples of Parameters of Observed Laws, Zavodskaya Laboratoriya, 1998, vol. 64, no. 3, pp. 61–72.
[2] Lemeshko, B.Yu. and Postovalov, S.N., Quality Management Methods, Nadezhnost’ i Kontrol’ Kachestva, 1999, no. 11, pp. 34–43.
[3] Lemeshko, B.Yu. and Postovalov, S.N., Applying Nonparametric Goodness-of-Fit Criteria for Testing Complex Hypotheses, Avtometriya, 2001, no. 2, pp. 88–102.
[4] R 50.1.037-2002 Rekomendatsii po standartizatsii. Prikladnaya statistika. Pravila proverki soglasiya opytnogo raspredeleniya s teoreticheskim. Ch. II. Neparametricheskie kriterii (Standartization Recommendations. Applied Statistics. Rules for Testing the Fit of Sample Distribution with the Theoretical One. Part II. Nonparametric Criteria), Moscow: Standarty, 2002.
[5] Lemeshko, B.Yu. and Maklakov, A.A., Nonparametric Criteria for Testing Complex Goodness-of-Fit Criteria for Distributions of the Exponential Family, Avtometriya, 2004, no. 3, pp. 3–20.
[6] Denisov, V.I., Eger, K.-H., Lemeshko, B.Yu., and Tsoy, E.B., Design of Experiments and Statistical Analysis for Grouped Observations, Novosibirsk: NSTU, 2004.
[7] Lemeshko, B.Yu. and Lemeshko, S.B., Distribution Models for Nonparametric Goodness-of-Fit Criteria for Testing Complex Hypotheses with Maximal Likelihood Estimates. I, Izmerit. Tekh., 2009, no. 6, pp. 6–11.
[8] Lemeshko, B.Yu. and Lemeshko, S.B., Distribution Models for Nonparametric Goodness-of-Fit Criteria for Testing Complex Hypotheses with Maximal Likelihood Estimates. II, Izmerit. Tekh., 2009, no. 8, pp. 17–26.
[9] Denisov, V.I., Lemeshko, B.Yu., and Tsoi, E.B., Optimal’noe gruppirovanie, otsenka parametrov i planirovanie regressionnykh eksperimentov (Optimal Grouping, Parameter Estimation, and Planning Regression Experiments), 2 vols., Novosibirsk: Novosibirsk. Gos. Tekhn. Univ., 1993.
[10] Lemeshko, B.Yu. and Postovalov, S.N., On the Dependence of Limit Distributions of Pearson’s \(\chi\)2 Statistics and Likelihood Ratios on the Method of Data Grouping, Zavodskaya Laboratoriya, 1998, vol. 64, no. 5, pp. 56–63.
[11] Lemeshko, B.Yu. and Chimitova, E.V., Maximizing the Power of \(\chi\) 2-type Criteria, Dokl. Sib. Otd. Akad. Nauk Vyssh. Shkol., 2000, no. 2, pp. 53–61.
[12] Lemeshko, B.Yu., Postovalov, S.N., and Chimitova, E.V., On Statistics and Power Distributions of Nikulin’s Criterion of \(\chi\) 2 Type, Zavodskaya Laboratoriya. Diagnostika Materialov, 2001, vol. 67, no. 3, pp. 52–58.
[13] Lemeshko, B.Yu. and Chimitova, E.V., On Errors andWrong Actions in Applications of Goodness-of-Fit Criteria of \(\chi\) 2 Type, Izmerit. Tekh., 2002, no. 6, pp. 5–11.
[14] R 50.1.033-2001 Rekomendatsii po standartizatsii. Prikladnaya statistika. Pravila proverki soglasiya opytnogo raspredeleniya s teoreticheskim. Ch. I. Kriterii tipa khi-kvadrat (Standartization Recommendation. Applied Statistics. Rules for Testing the Fit of Sample Distribution with the Theoretical One. Part I. Criteria of the Chi-Squared Type), Moscow: Standarty, 2002.
[15] Lemeshko, B.Yu., Lemeshko, S.B., and Postovalov, S.N., The Power of Goodness-of-Fit Criteria for Close Alternatives, Izmerit. Tekh., 2007, no. 2, pp. 22–27. · Zbl 1223.62064
[16] Lemeshko, B.Yu., Lemeshko, S.B., and Postovalov, S.N., Comparative Analysis of the Power of Goodness-of-Fit Criteria for Close Competing Hypotheses. I. Testing Simple Hypotheses, Sib. Zh. Industrial’noi Mat., 2008, vol. 11, no. 2(34), pp. 96–111. · Zbl 1223.62064
[17] Lemeshko, B.Yu., Lemeshko, S.B., and Postovalov, S.N., Comparative Analysis of the Power of Goodness-of-Fit Criteria for Close Competing Hypotheses. II. Testing Complex Hypotheses, Sib. Zh. Industrial’noi Mat., 2008, vol. 11, no. 4(36), pp. 78–93. · Zbl 1223.62065
[18] Lemeshko, B.Yu., On Estimating Distribution Parameters and Hypothesis Testing with Censored Samples, Metody Menedzh. Kachestva, 2001, no. 4, pp. 32–38.
[19] Lemeshko, B.Yu., Gil’debrant, S.Ya., and Postovalov, S.N., On Estimating Reliability Parameters with Censored Samples, Zavodskaya Laboratoriya. Diagnostika Materialov, 2001, vol. 67, no. 1, pp. 52–64.
[20] Lemeshko, B.Yu. and Pomadin, S.S., Testing Hypotheses on Expectations and Variances in Metrology and Quality Control Problems for Probabilistic Laws Deviating from the Normal Law, Metrologiya, 2004, no. 3, pp. 3–15.
[21] Lemeshko, B.Yu. and Lemeshko, S.B., On the Stability and Power of Mean Uniformity Criteria, Izmerit. Tekh., 2008, no. 9, pp. 23–28. · Zbl 1223.62064
[22] Lemeshko, S.B., The AbbĂ© Independence Criterion in Case Normality Assumptions are Violated, Izmerit. Tekh., 2006, no. 10, pp. 9–14.
[23] Lemeshko, B.Yu. and Mirkin, E.P., Bartlett’s and Cochrane’s Criteria in Measurement Problems for Probabilistic Laws Deviating from Normal, Izmerit. Tekh., 2004, no. 10, pp. 10–16.
[24] Lemeshko, B.Yu. and Lemeshko, S.B., Extending the Applicability of Grubbs Type Criteria Used for Outlier Rejection, Izmerit. Tekh., 2005, no. 6, pp. 13–19.
[25] Lemeshko, B.Yu. and Lemeshko, S.B., On the Convergence of Statistics and Power Distributions for Uniformity Criteria of Smirnov and Lehmann-Rosenblatt, Izmerit. Tekh., 2005, no. 12, pp. 9–14.
[26] Lemeshko, B.Yu. and Ogurtsov, D.V., Statistical Modeling as an Efficient Tool for Studying Distribution Laws of Functions of Random Values, Metrologiya, 2007, no. 5, pp. 3–13.
[27] Lemeshko, B.Yu. and Lemeshko, S.B., A Comparative Analysis of Criteria for a distribution’s Deviation from the Normal Law, Metrologiya, 2005, no. 2, pp. 3–24.
[28] Lemeshko, B.Yu. and Rogozhnikov, A.P., Studies of Power and Specific Features of Several Normality Criteria, Metrologiya, 2009, no. 4, pp. 3–24.
[29] Lemeshko, B.Yu. and Pomadin, S.S., Correlation Analysis of Samples of Multidimensional Random Variables When Normality Assumptions Are Violated, Sib. Zh. Industrial’noi Mat., 2002, vol. 5, no. 3, pp. 115–130. · Zbl 1075.62520
[30] Kolmogoroff, A.N., Sulla determinazione empirica di una legge di distribuzione, Giornale dell’ Istituto Italiano degly Attuari, 1933, vol. 4, no. 1, pp. 83–91. · JFM 59.1166.03
[31] Bol’shev, L.N., Asymptotic Pearson Transformations, Theor. Prob. Appl., 1963, vol. 8, no. 2, pp. 129–155. · Zbl 0125.09103 · doi:10.1137/1108012
[32] Bol’shev, L.N., Teoriya veroyatnostei i matematicheskaya statistika (Probability Theory and Mathematical Statistics), in Collected works, Yu.V. Prokhorov, Ed., Moscow: Nauka, 1987.
[33] Bol’shev, L.N. and Smirnov, N.V., Tablitsy matematicheskoi statistiki (Tables in Mathemetical Statistics), Moscow: Nauka, 1983.
[34] Anderson, T.W. and Darling, D.A., Asymptotic Theory of Certain ”Goodness of Fit” Criteria Based on Stochastic Processes, Ann. Math. Stat., 1952, vol. 23, pp. 193–212. · Zbl 0048.11301 · doi:10.1214/aoms/1177729437
[35] Anderson, T.W. and Darling, D.A., A Test of Goodness of Fit, J. Am. Stat. Ass., 1954, vol. 29, pp. 765–769. · Zbl 0059.13302 · doi:10.1080/01621459.1954.10501232
[36] Kac, M., Kiefer, J., and Wolfowitz, J., On Tests of Normality and Other Tests of Goodness of Fit Based on Distance Methods, Ann. Math. Stat., 1955, vol. 26, pp. 189–211. · Zbl 0066.12301 · doi:10.1214/aoms/1177728538
[37] Darling, D.A., The Cramer-Smirnov Test in the Parametric Case, Ann. Math. Stat., 1955, vol. 26, pp. 1–20. · Zbl 0064.13701 · doi:10.1214/aoms/1177728589
[38] Darling, D.A., The Cramer-Smirnov Test in the Parametric Case, Ann. Math. Stat., 1957, vol. 28, pp. 823–838. · Zbl 0082.13602 · doi:10.1214/aoms/1177706788
[39] Lilliefors, H.W., On the Kolmogorov-Smirnov Test for Normality with Mean and Variance Unknown, J. Am. Stat. Ass., 1967, vol. 62, pp. 399–402. · doi:10.1080/01621459.1967.10482916
[40] Lilliefors, H.W., On the Kolmogorov-Smirnov Test for the Exponential Distribution with Mean Unknown, J. Am. Stat. Ass., 1969, vol. 64, pp. 387–389. · doi:10.1080/01621459.1969.10500983
[41] Durbin, J., Weak Convergence of the Sample Distribution Function When Parameters Are Estimated, Ann. Stat., 1973, vol. 1, pp. 279–290. · Zbl 0256.62021 · doi:10.1214/aos/1176342365
[42] Durbin, J., Kolmogorov-Smirnov Tests When Parameters Are Estimated with Applications to Tests of Exponentiality and Tests of Spacings, Biometrika, 1975, vol. 62, pp. 5–22. · Zbl 0297.62027 · doi:10.1093/biomet/62.1.5
[43] Durbin, J., Kolmogorov-Smirnov Test When Parameters Are Estimated, Lect. Notes Math., 1976, vol. 566, pp. 33–44. · Zbl 0356.62021 · doi:10.1007/BFb0096877
[44] Gikhman, I.I., Several Notes on the Feasibility of A.N. Kolmogorov’s Criterion, Dokl. Akad. Nauk SSSR, 1953, vol. 91(4), pp. 715–718.
[45] Gihman, I.I., On the Empirical Distribution Function in the Case of Grouping Data, in Selected Translations in Mathematical Statistics and Probability, Providence: Am. Math. Soc., 1961, vol. 1, pp. 77–81.
[46] Martynov, G.V., Kriterii omega-kvadrat (Omega-squared Criteria), Moscow: Nauka, 1978. · Zbl 0949.62531
[47] Dzhaparidze, K.O. and Nikulin, M.S., Probability Distribution of the Kolmogorov and Omega-Square Statistics for Continuous Distributions with Shift and Scale Parameters, J. Soviet Math., 1982, vol. 20, pp. 2147–2163. · Zbl 0493.62043 · doi:10.1007/BF01239992
[48] Nikulin, M.S., Gihman and Goodness-of-Fit Tests for Grouped Data, Math. Reports Acad. Sci. Royal Soc. of Canada, 1992, vol. 14, no. 4, pp. 151–156. · Zbl 0768.62032
[49] Nikulin, M.S., A Variant of the Generalized Omega-Square Statistic, J. Soviet Math., 1992, vol. 61, no. 4, pp. 1896–1900. · Zbl 0784.62013 · doi:10.1007/BF01362800
[50] Pearson, E.S. and Hartley, H.O., Biometrica Tables for Statistics, Cambridge: Cambridge Univ. Press, 1972, vol. 2.
[51] Stephens, M.A., Use of Kolmogorov-Smirnov, Cramer von Mises and Related Statistics without Extensive Table, J. R. Stat. Soc., 1970, vol. 32, pp. 115–122. · Zbl 0197.44902
[52] Stephens, M.A., EDF Statistics for Goodness of Fit and Some Comparisons, J. Am. Stat. Ass., 1974, vol. 69, pp. 730–737. · doi:10.1080/01621459.1974.10480196
[53] Chandra, M., Singpurwalla, N.D., and Stephens, M.A., Statistics for Test of Fit for the Extrem-Value and Weibull Distribution, J. Am. Stat. Ass., 1981, vol. 76(375), pp. 729–731.
[54] Tyurin, Yu.N., On the Limit Distribution for Kolmogorov-Smirnov Statistics in the Case of a Complex Hypothesis, Izv. Akad. Nauk SSSR, Ser. Mat., 1984, vol. 48, no. 6, pp. 1314–1343. · Zbl 0571.62009
[55] Tyurin, Yu.N. and Savvushkina, N.E., Goodness-of-Fit Criteria for the Weibull-Gnedenko Distribution, Izv. Akad. Nauk SSSR, Ser. Tekhn. Kibern., 1984, no. 3, pp. 109–112. · Zbl 0645.62023
[56] Savvushkina, N.E., The Kolmogorov-Smirnov Criterion for Logistic and Gamma Distributions, Proc. of VNII Sistem. Issled., 1990, no. 8, pp. 50–56.
[57] Bagdonavicius, V., Clerjaud, L., and Nikulin, M.S., Accelerated Life Testing When the Hazard Rate Function Has Cup Shape, in Mathematical Methods in Survival Analysis, Reliability and Quality of Life, Huber, C., Limnios, N., Mesbah, M., and Nikulin, M., Eds., London: Wiley, 2008, pp. 203–215.
[58] Lemeshko, S.B., Nikulin, M.S., and Saaidia, N., Simulation of the Statistics Distributions and Power of the Goodness-of-Fit Tests in Composite Hypotheses Testing Rather Inverse Gaussian Distribution, Proc. 6th St. Petersburg Workshop on Simulation, St. Petersburg, 2009, vol. 1, pp. 323–328.
[59] Lemeshko, B.Yu., Lemeshko, S.B., Chimitova, E.V., and Postovalov, S.N., Computer Methods for Investigating Statistical Regularities in Problems of Statistical Data Analysis and Reliability, Proc. 6th Int. Conf. ”Mathematical Methods in Reliability,” Moscow, 2009, pp. 418–422.
[60] Greenwood, P.E. and Nikulin, M.S., A Guide to Chi-Squared Testing, New York: Wiley, Inc., 1996.
[61] Nikulin, M.S., On the Chi-Squared Criterion for Continuous Distributions, Theor. Prob. Appl., 1973, vol. XVIII, no. 3, pp. 675–676.
[62] Nikulin, M.S., The Chi-Squared Criterion for Continuous Distributions with Shift and Scale Parameters, Theor. Prob. Appl., 1973, vol. XVIII, no. 3, pp. 583–591.
[63] Rao, K.C. and Robson, D.S., A Chi-Squared Statistic for Goodness-of-Fit Tests within the Exponential Family, Commun. Stat., 1974, vol. 3, pp. 1139–1153. · Zbl 0294.62057 · doi:10.1080/03610927408827216
[64] Lemeshko, B.Yu. and Postovalov, S.N., Computer Technologies in Data Mining and Studying Statistical Patterns. Textbook, Novosibirsk: Novosibirsk. Gos. Tekh. Univ., 2004.
[65] Lemeshko, B.Yu., Postovalov, S.N., and Lemeshko, S.B., Komp’yuternye tekhnologii analiza dannykh i issledovaniya statisticheskikh zakonomernostei. Metodicheskie ukazaniya k vypolneniyu laboratornykh rabot (Computer Technologies in Data Mining and Studying Statistical Patterns. Guidelines for Laboratory Works), Novosibirsk: Novosibirsk. Gos. Tekh. Univ., 2007.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.