Abstract
In this paper an analogue of the formulas [D. M. Chibisov,Teor. Veroyatn. Primen.,30, 269–288 (1985);Izv. Akad. Nauk UzSSR,6, 23–30 (1982)] for the difference between the power of a given asymptotically efficient test and that of the most powerful test is justified for one-sample L-and R-tests, i.e., tests based on linear combinations of order statistics and linear rank statistics. This formula directly yields the Hodges-Lehmann deficiency of corresponding tests. A general theorem is stated which is applied to L-and R-tests. The explicit expressions given by this formula for L- and R-tests are also presented. The expression related to R-tests agrees with the one obtained in [W. Albers, P. J. Bickel, and W. R. Van Zwet,Ann. Statist.,4, 108–156 (1976);6, 1170–1171 (1978)]. We present here a nontechnical (heuristic) proof of these results.
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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, “Power and deficiency of asymptotically optimal tests,”Teor. Veroyatn. Primen.,30, 269–288, (1985).
D. M. Chibisov, and W. R. Van Zwet, “On the Edgeworth expansion for the distribution of the logarithm of likelihood ratio,”Teor. Veroyatn. Primen.,29, 417–439 (1984).
J. L. Jr. Hodges and E. L. Lehmann, “Deficiency,”Ann. Math. Statist.,41, 783–801 (1970).
R. Helmers, “Edgeworth expansions for linear combinations of order statistics with smooth weight functions,”Math. Centr., Rep. Sw.44/76, Amsterdam (1976).
D. M. Chibisov, “Asymptotic expansions in test procedures,”Izv. Akad. Nauk UzSSR,6, 23–30 (1982).
D. M. Chibisov, “Asymptotic expansions and deficiencies of tests,” in:Proceedings of the International Congress of Mathematicians, Vol. 2, Warszawa, pp. 1063–1079.
J. Pfanzagl, “Asymptotic expansions in parametric statistical theory,” in:Development in Statistics, Vol. 3. Academic Press, New York (1980), pp. 1–97.
J. Pfanzagl, “First order efficiency implies second order efficiency,” in:Contributions to Statistics, Jaroslaw Hájek Memorial Volume, J. Jureckova, ed., Academia, Prague (1979), pp. 167–196.
D. M. Chibisov,Asymptotic Expansions for Some Asymptotically Optimal Tests, Vol. 2, Charles Univ., Prague (1974), pp. 37–68.
V. K. Malinovskii, “On power function and deficiency of tests in the case of Markov observations,”Teor. Veroyatn. Primen.,34, No. 3, 491–504 (1989).
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. Statist.,4, 108–156 (1976); Correction6, 1170–1171 (1978).
J. Hájek and Z. Sidak,Theory of Rank Tests, Academia, Prague (1967).
L. Schmetterer,Einfuhrung in die Mathematische Statistik, Springer-Verlag, Wien-New York (1966).
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Supported by the Russian Foundation for Fundamental Research (grant No. 93-011-1446).
Proceedings of the XVI Seminar on Stability Problems for Stochastic Models, Part II, Eger, Hungary, 1994.
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Bening, V.E. A formula for deficiency (L-andR-tests). J Math Sci 78, 18–27 (1996). https://doi.org/10.1007/BF02367951
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DOI: https://doi.org/10.1007/BF02367951