Abstract
One computes the local asymptotic Chernoff and Hodges—Lehmann efficiencies of linear rank statistics for the testing of symmetry under certain regularity conditions, imposed on the labels and on the distribution of the observations under the alternative. It is revealed that these efficiencies coincide with the previously known Pitman and local Bahadur efficiencies.
Similar content being viewed by others
References
E. Pitman, Lecture Notes on Nonparametric Statistical Inference. Mimeographed. Columbia University, New York (1949).
M. Kendall and A. Stuart, The Advanced Theory of Statistics, Vol. 2: Inference and Relationship, Hafner, New York (1973).
J. W. Pratt and J. D. Gibbons, Concepts of Nonparametric Theory, Springer, New York (1981).
R. R. Bahadur, "Rates of convergence of estimates and test statistics," Ann. Math. Statist.,38, No. 2, 303–324 (1967).
R. R. Bahadur, Some Limit Theorems in Statistics, SIAM, Philadelphia (1971).
J. L. Hodges, Jr. and E. L. Lehmann, "The efficiency of some nonparametric competitors of the t-test," Ann. Math. Statist.,27, No. 2, 324–335 (1956).
M. Raghavachari, "On the computation of Hodges—Lehmann efficiencies of test statistics," in: A Festschrift for Erich L. Lehmann, Wadsworth, Belmont, Calif. (1983), pp. 367–378.
H. Chernoff, "A measure of asymptotic efficiency for tests of a hypothesis based on sums of observations," Ann. Math. Statist.,23, No. 4, 493–507 (1952).
W. C. M. Kallenberg, "Chernoff efficiency and deficiency," Ann. Statist.,10, No. 2, 583–594 (1982).
H. Chernoff and I. R. Savage, "Asymptotic normality and efficiency of certain nonparametric test statistics," Ann. Math. Statist.,29, No. 4, 972–994 (1958).
I. N. Sanov, "On the probability of large deviations of random variables," Mat. Sb.,42 (84), No. 1, 11–44 (1957).
P. Groeneboom, J. Oosterhoff, and F. H. Ruymgaart, "Large deviation theorems for empirical probability measures," Ann. Probab.,7, No. 4, 553–586 (1979).
Ya. Yu. Nikitin, "Chernoff efficiency of the sign and Wilcoxon tests for testing the symmetry hypothesis," Zap. Nauchn.2 Sem. Leningr. Otd. Mat. Inst.,177, 101–107 (1989).
Ya. Yu. Nikitin, "On the Hodges—Lehmann efficiency of nonparametric goodness-of-fit and homogeneity tests," Teor. Veroyatnost. i Primenen.,32, No. 1, 82–91 (1987).
Ya. Yu. Nikitin, "Hodges—Lehmann efficiency of nonparametric tests," in: Probability Theory and Mathematical Statistics, Vol. II (Vilnius, 1985), VNU Sci. Press, Utrecht (1987), pp. 391–408.
Ya. Yu. Nikitin, "Local Chernoff efficiency of linear rank statistics," in: Abstracts of the Fifth Vilnius Conference on Mathematical Statistics and Probability, Vol. 2 (1989), pp. 49–50.
T. Khuttmansperger, Statistical Inferences Based on Ranks [in Russian], Moscow (1987).
P. Groeneboom, Large Deviations and Asymptotic Efficiencies, Mathematical Centre Tracts No. 118, Math. Centrum, Amsterdam (1980).
Nguyen van Ho, "Asymptotic efficiency in the Bahadur sense for the signed rank tests," in: Proc. Prague Symp. on Asymptotic Statistics. Vol. II (Prague, 1973), Charles University, Prague (1974).
T. Ledwina, "An expansion of the index of large deviations for linear rank statistics," Statist. Probab. Lett.,5, No. 4, 247–248 (1987).
V. M. Alekseev, V. M. Tikhomirov, and S. V. Fomin, Optimal Control, Consultants Bureau, New York (1987).
Additional information
Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 184, pp. 215–226, 1990.
Rights and permissions
About this article
Cite this article
Nikitin, Y.Y. Local Chernoff and Hodges-Lehmann efficiencies of linear rank tests for symmetry. J Math Sci 68, 551–559 (1994). https://doi.org/10.1007/BF01254281
Issue Date:
DOI: https://doi.org/10.1007/BF01254281