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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
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[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
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