Abstract
Asymptotic distributions of test statistics under alternatives are important from the point of view of their power properties. When the limiting distributions of test statistics are specified under the hypothesis in a certain sense, LeCam's third lemma ([4], Chapter 6) enables one to obtain their limiting distributions under close alternatives. In this paper we generalize LeCam's third lemma by using the rate of convergence in the case of asymptotically efficient test statistics. A general lemma is proved which is specified to linear combinations of order statistics (L-statistics) and linear rank statistics (R-statistics). Edgeworth-type asymptotic expansions for these statistics under alternatives are considered in [3].
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References
P. J. Bickel, D. M. Chibisov, and W. R. Van Zwet, “On efficiency of first and second order,”Internat. Statist. Rev.,49, 169–175 (1981).
D. M. Chibisov, “Asymptotic expansions and deficiencies of tests,” in:Proceedings of the International Congress of Mathematicians,2, Warszawa, 1063–1079 (1983).
V. E. Bening, “A method for obtaining asymptotic expansions based on properties of the likelihood ratio,”Bull. Moscow Univ.,15, No. 2, 36–44 (1994).
J. Hajek and Z. Sidak,Theory of Rank Tests, Academia, Prague, (1967).
D. M. Chibisov and W. R. Van Zwet, “On the Edgeworth expansion for the distribution of the logarithm of the lekilihood ratio,”Probab. Theory Appl.,29, No. 3, 417–439 (1984).
V. V. Petrov,Sums of Independent Random Variables, Springer-Verlag, Berlin-New York (1975).
R. Helmers, “The order of the normal approximation for linear combinations of order statistics with smooth weight functions,”Ann. Prob.,5, No. 6, 940–963 (1974).
S. V. Nagaev, “Some limit theorem for large deviations,”Probab. Theory Appl.,10, No. 2, 231–254 (1965).
W. Albers, P. J. Bickel, and W. R. Van Zwet, “Asymptotic expansion for the power of distribution free tests in the one-sample problem,”Ann. Stat.,4, 108–156 (1976).
S. M. Stigler, “Linear functions of order statistics with smooth weight functions,”Ann. Stat.,2, No. 4, 676–693 (1974).
H. Bergstrom and M. L. Puri, “Convergence and remainder terms in linear rank statistics,”Ann. Stat.,5, No. 4, 671–680 (1977).
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Supported by the Russian Foundation for Fundamental Research (grant No. 93-01-01446).
Proceedings of the XVI Seminar on Stability Problems for Stochastic Models, Part I, Eger, Hungary, 1994.
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Bening, V.E. On the rate of convergence ofL- andR-statistics under alternatives. J Math Sci 76, 2227–2240 (1995). https://doi.org/10.1007/BF02362693
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DOI: https://doi.org/10.1007/BF02362693