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An Extension of the Csörgo-Horváth Functional Limit Theorem and Its Applications to Changepoint Problems

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  • Ferger, D.

Abstract

Consider a triangular array Xn1, ..., Xnn, n[set membership, variant] 1, of rowwise independent random elements with values in a measurable space. Suppose there exists [theta] [set membership, variant] [0, 1)such that Xn1, ..., Xn[n[theta] ] have distribution [nu]1 and Xn[n[theta]] + 1(n), ..., Xnn have distribution [nu]2. Csörgo and Horváth derived an invariance principle for a one-time parameter process, which is the foundation of a test for H0: [theta] = 0 versus H1: [theta] [set membership, variant](0, 1). We are interested in the more complex test problem H0: [theta] [set membership, variant] [Theta]0 versus H1,: [theta] [set membership, variant] [Theta]0, where [Theta]0[subset, double equals](0, 1). To treat this new situation, we extend the Csörgo-Horváth result in proving a functional limit theorem for a suitable two-time parameter process. We briefly sketch several applications of our result. Especially, the power of the Csörgo-Horváth test is investigated in detail.

Suggested Citation

  • Ferger, D., 1994. "An Extension of the Csörgo-Horváth Functional Limit Theorem and Its Applications to Changepoint Problems," Journal of Multivariate Analysis, Elsevier, vol. 51(2), pages 338-351, November.
    • Handle: RePEc:eee:jmvana:v:51:y:1994:i:2:p:338-351
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    Cited by:

    1. Gombay, Edit, 2001. "U-Statistics for Change under Alternatives," Journal of Multivariate Analysis, Elsevier, vol. 78(1), pages 139-158, July.
    2. Steland Ansgar, 2003. "Jump-preserving monitoring of dependent time series using pilot estimators," Statistics & Risk Modeling, De Gruyter, vol. 21(4/2003), pages 343-366, April.
    3. Zhang, Hanqin, 2000. "On a weighted embedding for generalized pontograms," Stochastic Processes and their Applications, Elsevier, vol. 88(2), pages 213-224, August.
    4. Ansgar Steland, 2005. "Random Walks with Drift – A Sequential Approach," Journal of Time Series Analysis, Wiley Blackwell, vol. 26(6), pages 917-942, November.
    5. Steland Ansgar, 2003. "NP-Optimal Kernels for Nonparametric Sequential Detection Rules," Stochastics and Quality Control, De Gruyter, vol. 18(2), pages 149-163, January.

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