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Limit theorems for rank statistics. (English) Zbl 0933.62039
Summary: The purpose of this paper is to extend the weak asymptotic results for weighted partial sums of i.i.d. random variables to weighted partial sums of rank scores. These results then suggest various test procedures for the change point problem. The crucial tools in the proofs are martingale properties of a class of two-sample rank statistics and J. Hájek’s [Ann. Math. Stat. 32, 506–523 (1961; Zbl 0107.13404)] results on simple linear rank statistics.

##### MSC:
 62G20 Asymptotic properties of nonparametric inference 62E20 Asymptotic distribution theory in statistics 60F17 Functional limit theorems; invariance principles
##### Keywords:
rank statistics; weighted maxima; location model
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##### References:
  Chen, X., Inference in a simple change-point problem, Scientia sinica, A31, 654-667, (1988) · Zbl 0691.62025  Chow, Y.S.; Teicher, H., Probability theory, (1988), Springer New York, 1988  Csörgő, M.; Horváth, L., Weighted approximations in probability and statistics, (1993), Wiley New York  Csörgő, M.; Horváth, L., A note on the change-point problem for angular data, Statist. probab. lett., (1996), to appear, in:  Csörgő, M.; Horváth, L., Limit theorems in change point analysis, (1996), Wiley New York  Gombay, E.; Hušková, M., Rank based estimators of the change point, (1995), in preparation  Hájek, J., Some extensions of the Wald-wolfowitz-Noether theorem, Ann. math. statist., 32, 506-523, (1961) · Zbl 0107.13404  Höglund, Sampling from a finite population. A remainder term estimates population, Scand. J. statist., 5, 69-71, (1978) · Zbl 0382.60028  Lombard, E., Rank tests for change point problem, Biometrika, 74, 615-624, (1987) · Zbl 0628.62047  Praagman, J., Bahadur efficiency of tests for a shift in location of normal population, (), 135-161  Sen, P.K., Invariance principles for rank statistics revisited, Sankhya ser. A, 40, 215-236, (1978) · Zbl 0439.62006  Sen, P.K., Sequential nonparametrics, (1981), Wiley Chichester · Zbl 0583.62074  Wolfe, D.A.; Schechtman, E., Nonparametric procedures for the change point problem, J. statist. plan. infer., 9, 389-396, (1984) · Zbl 0561.62039  Yao, Y.-C., On asymptotic behavior of a class of nonparametric tests, Statistics probab. lett., 9, 173-177, (1990) · Zbl 0686.62030
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