The application of the nonparametric Anderson–Darling, Cramer–Mises–Smirnov, Kuiper, Watson, Kolmogorov, and Zhang goodness-of-fit tests in verification of simple and composite hypotheses is considered. Based on an investigation of the power, it is shown for the first time that there exist pairs of competing hypotheses which these tests are not able to distinguish in the case of small sample sizes n and type 1 error probabilities. It is shown that the reason for this lies in the bias of the tests in corresponding situations.
Similar content being viewed by others
References
B. Yu. Lemeshko, Nonparametric Goodness-of-Fit Tests: Handbook on Applications, NITs INFRA-M, Moscow (2014), DOI: 10.12737/11873.
L. N. Bol’shev and N. V. Smirnov, Tables of Mathematical Statistics, Nauka, Moscow (1983).
N. H. Kuiper, “Tests concerning random points on a circle,” Proc. Konikl. Nederl. Akad. Van Wettenschappen, Ser. A, 63, 38–47 (1960).
M. A. Stephens, “EDF statistics for goodness of fit and some comparisons,” J. Amer. Stat. Assoc., 69, No. 347, 730–737 (1974).
B. Yu. Lemeshko and A. A. Gorbunova, “On the application and power of the Kuiper, Watson, and Zhang nonparametric goodness-of-fit tests,” Izmer. Tekhn., No. 5, 3–9 (2013).
G. S. Watson, “Goodness-of-fit tests on a circle,” Biometrika, 48, No.1–2, 109–114 (1961).
G. S. Watson, “Goodness-of-fit tests on a circle,” Biometrika, 49, No. 1–2, 57–63 (1962).
T. W. Anderson and D. A. Darling, “Asymptotic theory of certain ‘goodness-of-fit’ criteria based on stochastic processes,” Ann. Math. Stat., 23, 193–212 (1952).
T. W. Anderson and D. A. Darling, “A test of goodness of fit,” J. Amer. Stat. Assoc., 29, 765–769 (1954).
J. Zhang, Powerful Goodness-of-fit and Multi-sample Tests: PhD Thesis, York University, Toronto (2001).
J. Zhang, “Powerful goodness-of-fit tests based on the likelihood ratio,” J. Roy. Stat. Soc.: Ser. B, 64, No. 2, 281–294 (2002).
M. Kac, J. Kiefer, and J. Wolfowitz, “On tests of normality and other tests of goodness of fit based on distance methods,” Ann. Math. Stat., 26, 189–211 (1955).
B. Yu. Lemeshko, S. B. Lemeshko, and S. N. Postovalov, “Statistic distribution models for some nonparametric goodness-of-fit tests in testing composite hypotheses,” Comm. Stat. Theory and Methods, 39, No. 3, 460–471 (2010).
B. Yu. Lemeshko and S. B. Lemeshko, “Models of statistic distributions of nonparametric goodness-of-fit tests in composite hypotheses testing for double exponential law cases,” Comm. Stat. Theory and Methods, 40, No. 16, 2879–2892 (2011).
B. Yu. Lemeshko and S. B. Lemeshko, “Models of distributions of statistics of nonparametric goodness-of-fit tests in verification of composite hypotheses with the use of maximum likelihood estimators. Part I,” Izmer. Tekhn., No. 6, 3–11 (2009).
B. Yu. Lemeshko and S. B. Lemeshko, “Models of distributions of statistics of nonparametric goodness-of-fit tests in verification of composite hypotheses with the use of maximum likelihood estimators. Part II,” Izmer. Tekhn., No. 8, 17–26 (2009).
B. Yu. Lemeshko, A. A. Gorbunova, S. B. Lemeshko, and A. R. Rogozhnikov, “Solving problems of using some nonparametric goodness-of-fit tests,” Optoelectr., Instrum. Data Proces., 50, 21–35 (2014).
B. Yu. Lemeshko and A. A. Gorbunova, “Use of Kuiper and Watson nonparametric goodness-of-fit tests in verification of composite hypotheses,” Izmer. Tekhn., No. 9, 14–21 (2013).
B. Yu. Lemeshko and S. B. Lemeshko, “A comparative analysis of tests for verification of deviation of a distribution from a normal law,” Metrologiya, No. 2, 3–24 (2005).
B. Yu. Lemeshko and A. P. Rogozhnikov, “Investigation of features and power of certain normality tests,” Metrologiya, No. 4, 3–24 (2009).
B. Yu. Lemeshko, Tests for Verification of Deviation of a Distribution from a Normal Law: Handbook on Applications, INFRA-M, Moscow (2015), DOI: 10.12737/6086.
B. Yu. Lemeshko, P. Yu. Blinov, Tests for Verification of Deviation of a Distribution from a Normal Law: Handbook on Applications, INFRA-M, Moscow (2015), DOI: 10.12737/11304.
GOST R 8.736–2011, Direct Repeated Measurements. Methods of Processing the Results of Measurements. Basic Assumptions.
S. S. Antsyferov, M. S. Afanas’ev, and K. E. Rusanov, Processing the Results of Measurements: Teach. Aid, Izd. IKAR, Moscow (2014).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Izmeritel’naya Tekhnika, No. 5, pp. 16–20, May, 2016.
Rights and permissions
About this article
Cite this article
Lemeshko, B.Y., Blinov, P.Y. & Lemeshko, S.B. Bias of Nonparametric Goodness-of-Fit Tests Relative to Certain Pairs of Competing Hypotheses. Meas Tech 59, 468–475 (2016). https://doi.org/10.1007/s11018-016-0992-3
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11018-016-0992-3