Abstract
For the Kolmogorov and omega-square tests, strong asymptotics for large deviations of type II error probabilities are obtained in the case of “least favorable alternatives.” Using these asymptotics, type II error probabilities for any sequence of alternatives can readily be estimated. The proofs are based on an exact asymptotic of large deviation probabilities for Gaussian measures in a Hilbert space and on a theorem on large deviation probabilities for sums of independent random vectors in a Banach space. Bibliography: 22 titles.
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REFERENCES
I. G. Abrahamson, “Exact Bahadur efficiencies for Kolmogorov-Smirnov and Kuiper one and two-sample statistics,” Ann. Math. Statist., 38, 1475–1490 (1967).
V. Yu. Bentkus, “On large deviations in Banach spaces,” Teor. Veroyatn. Primen., 31, 710–716 (1986).
A. A. Borovkov and A. A. Mogulskii, “Large deviations and the statistical principle of invariance,” Teor. Veroyatn. Primen., 37, 11–18 (1992).
A. A. Borovkov and N. M. Sycheva, “On asymptotically optimal nonparametric tests,” Teor. Veroyatn. Primen., 13, 385–418 (1968).
M. S. Ermakov, “Asymptotic minimaxity of tests of the Kolmogorov and omega-square types,” Teor. Veroyatn. Primen., 40, 54–67 (1995).
M. S. Ermakov, “Large deviations of empirical measures and statistical tests,” Zap. Nauchn. Semin. POMI, 207, 37–60 (1993).
M. S. Ermakov, “Minimax testing of a nonparametric hypothesis against nonparametric sets of alternatives,” in: Probability Theory Mathematical Statistics. Fifth Vilnius Intern. Conf., Mokslas et al. (eds.), 1 (1990), pp. 175–184.
M. S. Ermakov, “On asymptotic minimaxity of rank tests,” Statist. Probab. Lett., 15, 191–196 (1992).
M. S. Ermakov, “Asymptotic minimaxity of chi-square tests,” Teor. Veroyatn. Primen., 42, 668–695 (1997).
M. S. Ermakov, “On asymptotic minimaxity of Kolmogorov and omega-square tests,” Statist. Probab. Lett., 30, 227–233 (1996).
M. S. Ermakov, “Minimax nonparametric hypothesis testing of the density of a distribution,” Teor. Veroyatn. Primen., 39, 488–512 (1994).
M. S. Ermakov, “On large deviation probabilities in a Banach space,” Zap. Nauchn. Semin. POMI, 278, 40–57 (2001).
G. G. Gregory, “On efficiency and optimality of quadratic tests,” Ann. Statist., 8, 116–132 (1980).
C. R. Hwang, “Gaussian measure of large balls in a Hilbert space,” Proc. Amer. Math. Soc., 78, 107–110 (1980); 94, 188 (1985).
Yu. I. Ingster, “Asymptotically minimax hypothesis testing for nonparametric alternatives. I, II, III,” Math. Meth. Statist., 2, 85–114, 171–189, 249–268 (1993).
J. Komlos, P. Major, and G. Tusnady, “An approximation of partial sums of independent RV’s and the sample DF,” Z. Wahrscheinlichtskeittheorie verw. Geb., 32, 111–131 (1975).
A. A. Mogulskii, “One remark on large deviations of omega-square statistics,” Teor. Veroyatn. Primen., 22, 170–175 (1977).
G. A. Nesenko and Yu. N. Turin, “Asymptotic of the Kolmogorov statistic for parametric families,” Dokl. Akad. Nauk USSR, 239, 1292–1294 (1978).
L. V. Osipov, “On large deviation probabilities for independent random vectors,” Teor. Veroyatn. Primen., 23, 510–525 (1978).
V. I. Paulauskas, “On rate of convergence in the central limit theorem in a Banach space,” Teor. Veroyatn. Primen., 21, 775–791 (1976).
V. I. Piterbarg, Asymptotic Methods in the Theory of Gaussian Random Processes and Fields [in Russian], Moskow (1988).
V. M. Zolotarev, “On a probability problem,” Teor. Veroyatn. Primen., 6, 219–222 (1961).
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 298, 2003, pp. 80–110.
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Ermakov, M.S. Probabilities of Large Deviations of Type II Errors for Tests of Kolmogorov and Omega-Square Types. J Math Sci 128, 2538–2555 (2005). https://doi.org/10.1007/s10958-005-0201-4
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DOI: https://doi.org/10.1007/s10958-005-0201-4