The application of the Pearson chi-square test for verification of the normality of a sample is discussed. Tables of percentage points and models for the limiting statistical distributions are constructed. The powers of the Pearson and Nikulin–Rao–Robson chi-square tests are estimated relative to various competing hypotheses. A comparative analysis of the powers of a set of normality tests is given.
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References
B. Yu. Lemeshko and S. B. Lemeshko, “Comparative analysis of tests for verifying deviation of a distribution from normal,” Metrologiya, No. 2, 3–24 (2005).
B. Yu. Lemeshko and A. P. Rogozhnikov, “Features and power of some tests of normality,” Metrologiya, No. 4, 3–24 (2009).
B. Yu. Lemeshko, S. B. Lemeshko, S. N. Postovalov, and E. V. Chimitova, Statistical Analysis of Data, Modelling, and Study of Probability Behavior. A Computerized Approach: Monograph, Izd NGTU, Novosibirsk (2011).
B. Yu. Lemeshko, Tests for Verification of Deviation of Distributions from Normal: Application Handbook, INFRA-M, Moscow (2014).
B. Yu. Lemeshko, Nonparametric Tests of Goodness-of-Fit: Application Handbook, INFRA-M, Moscow (2015).
H. W. Lilliefors, “On the Kolmogorov–Smirnov test for normality with mean and variance unknown,” J. Am. Statist. Assoc., 62, 399–402 (1967).
G. V. Martynov, Omega-Squared Tests, Nauka, Moscow (1978).
B. Yu. Lemeshko and A. A. Gorbunova, “Application and power of the nonparametric Kuiper, Watson, and Zhang tests of goodness-of-fit,” Izmer. Tekhn., No. 5, 3–9 (2013).
B. Yu. Lemeshko and A. A. Gorbunova, “Application of nonparametric Kuiper and Watson tests of goodness-of-fit for composite hypotheses,” Izmer. Tekhn., No. 9, 14–21 (2013).
B. Yu. Lemeshko, A. A. Gorbunova, S. B. Lemeshko, and A. P. Rogozhnikov, “Solving problems of using some nonparametric goodness-of-fit tests,” Optoelectr., Instrum., Data Proc., 50, No. 1, 21–35 (2014).
J. Zhang, Powerful Goodness-of-Fit and Multi-Sample Tests: Ph.D. Thesis, Toronto (2001).
M. S. Nikulin, “χ2 tests for continuous distributions with shift and scale parameters,” Teor. Veroyatn. Primen., XVIII, No. 3, 583–591 (1973).
M. S. Nikulin, “On χ2 tests for continuous distributions,” Teor. Veroyatn. Primen., XVIII, No. 3, 675–676 (1973).
K. C. Rao and D. S. Robson, “A chi-squared statistic for goodness-of-fit tests within the exponential family,”Commun. Stat., 3, 1139–1153 (1974).
H. Chernoff and E. L. Lehmann, “The use of maximum likelihood estimates in χ2 test for goodness of fit,” Ann. Math. Stat., 25, 579–586 (1954).
B. Yu. Lemeshko and S. N. Postovalov, “On the dependence of the limiting distributions of statistics for Pearson χ2 tests and the likelihood ratio on the means of grouping data,” Zavod. Lab., 64, No. 5, 56–63 (1998).
B. Yu. Lemeshko and E. V. Chimitova, “On the choice of the number of intervals in χ2-type goodness-of-fit tests,” Zavod. Lab. Diagn. Mater., 69, No. 1, 61–67 (2003).
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Translated from Izmeritel’naya Tekhnika, No. 6, pp. 3–9, June, 2015.
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Lemeshko, B.Y. Chi-Square-Type Tests for Verification of Normality. Meas Tech 58, 581–591 (2015). https://doi.org/10.1007/s11018-015-0759-2
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DOI: https://doi.org/10.1007/s11018-015-0759-2