×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Chen, X., Inference in a simple change-point problem, Scientia sinica, A31, 654-667, (1988) · Zbl 0691.62025
[2] Chow, Y.S.; Teicher, H., Probability theory, (1988), Springer New York, 1988
[3] Csörgő, M.; Horváth, L., Weighted approximations in probability and statistics, (1993), Wiley New York
[4] Csörgő, M.; Horváth, L., A note on the change-point problem for angular data, Statist. probab. lett., (1996), to appear, in:
[5] Csörgő, M.; Horváth, L., Limit theorems in change point analysis, (1996), Wiley New York
[6] Gombay, E.; Hušková, M., Rank based estimators of the change point, (1995), in preparation
[7] Hájek, J., Some extensions of the Wald-wolfowitz-Noether theorem, Ann. math. statist., 32, 506-523, (1961) · Zbl 0107.13404
[8] Höglund, Sampling from a finite population. A remainder term estimates population, Scand. J. statist., 5, 69-71, (1978) · Zbl 0382.60028
[9] Lombard, E., Rank tests for change point problem, Biometrika, 74, 615-624, (1987) · Zbl 0628.62047
[10] Praagman, J., Bahadur efficiency of tests for a shift in location of normal population, (), 135-161
[11] Sen, P.K., Invariance principles for rank statistics revisited, Sankhya ser. A, 40, 215-236, (1978) · Zbl 0439.62006
[12] Sen, P.K., Sequential nonparametrics, (1981), Wiley Chichester · Zbl 0583.62074
[13] Wolfe, D.A.; Schechtman, E., Nonparametric procedures for the change point problem, J. statist. plan. infer., 9, 389-396, (1984) · Zbl 0561.62039
[14] Yao, Y.-C., On asymptotic behavior of a class of nonparametric tests, Statistics probab. lett., 9, 173-177, (1990) · Zbl 0686.62030
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.