# zbMATH — the first resource for mathematics

Invariance principles for changepoint problems. (English) Zbl 0656.62031
The asymptotic behavior of certain processes $$U_ k$$, obtained from U- statistics, is studied. Motivation for doing this comes from testing the hypothesis $$H_ 0$$ that the independent random variables under consideration have the same distribution, versus the alternative $$H_ a$$ that at some moment in time $$\lambda$$, the sampling distribution has been changed (once through experiment only). Under $$H_ 0$$, weak convergence of $$n^{-3/2}U_{[(n+1)t]}/\sigma$$ to $$(1-t)W_ t+t\{W_ 1-W_ t\},$$ where $$\{W_ t\}_{0\leq t\leq 1}$$ is a Wiener process, is obtained as $$n\to \infty$$. The asymptotic behavior of $$\max_{1\leq k\leq n}U_ k[k(n-k+1)n]^{-1/2}/\sigma$$ under $$H_ 0$$ and as $$n\to \infty$$ is also analyzed.
Reviewer: W.Bryc

##### MSC:
 62F05 Asymptotic properties of parametric tests 62F03 Parametric hypothesis testing 60F17 Functional limit theorems; invariance principles
##### Keywords:
U-statistics; Wiener process
Full Text:
##### References:
 [1] Csörgő, M.; Csörgő, S.; Horváth, L.; Mason, D.M., Weighted empirical and quantile processes, Ann. probab., 14, 31-85, (1986) · Zbl 0589.60029 [2] Csörgő, M.; Horváth, L., Nonparametric tests for the changepoint problem, J. statist. plann. inference, 17, 1-9, (1987) · Zbl 0631.62056 [3] Csörgő, M.; Révész, P., () [4] Darling, D.; Erdős, P., A limit theorem for the maximum of normalized sums of independent random variables, Duke math. J., 23, 143-155, (1956) · Zbl 0070.13806 [5] Grams, W.F.; Serfling, R.J., Convergence rates for U-statistics and related statistics, Ann. statist., 1, 153-160, (1973) · Zbl 0322.62053 [6] Hall, P., On the invariance principle for U-statistics, Stochastic process. appl., 9, 163-174, (1979) · Zbl 0422.62019 [7] Hoeffding, W., The strong law of large numbers for U-statistics, (1961), University of North Carolina Institute of Statistics Mimeo Series 302 · Zbl 0211.20605 [8] Janson, S.; Wichura, M.J., Invariance principles for stochastic area and related stochastic integrals, Stochastic process. appl., 16, 71-84, (1983) · Zbl 0523.60014 [9] Pettitt, A.N., A non-parametric approach to the change-point problem, Appl. statist., 28, 126-135, (1979) · Zbl 0438.62037 [10] Pettitt, A.N., Some results on estimating a change-point using non-parametric type statistics, J. statist. comput. simul., 11, 261-272, (1980) · Zbl 0454.62039 [11] Sen, A.; Srivastava, M.S., On tests for detecting changes in Mean, Ann. statist., 3, 98-108, (1975) · Zbl 0305.62014 [12] Sen, P.K., Almost sure convergence of generalized U-statistics, Ann. probab., 5, 287-290, (1977) · Zbl 0362.60019 [13] Serfling, R.J., () [14] Shorack, G.R., Extension of the darling and erdős theorem on the maximum of normalized sums, Ann. probab., 7, 1092-1096, (1979) · Zbl 0421.60046 [15] Wolfe, D.A.; Schechtman, E., Nonparametric statistical procedures for the changepoint problem, J. statist. plann. inference, 9, 389-396, (1984) · Zbl 0561.62039
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.