Open Access
November 2011 Asymptotics of trimmed CUSUM statistics
István Berkes, Lajos Horváth, Johannes Schauer
Bernoulli 17(4): 1344-1367 (November 2011). DOI: 10.3150/10-BEJ318

Abstract

There is a wide literature on change point tests, but the case of variables with infinite variances is essentially unexplored. In this paper we address this problem by studying the asymptotic behavior of trimmed CUSUM statistics. We show that in a location model with i.i.d. errors in the domain of attraction of a stable law of parameter $0 < α < 2$, the appropriately trimmed CUSUM process converges weakly to a Brownian bridge. Thus, after moderate trimming, the classical method for detecting change points remains valid also for populations with infinite variance. We note that according to the classical theory, the partial sums of trimmed variables are generally not asymptotically normal and using random centering in the test statistics is crucial in the infinite variance case. We also show that the partial sums of truncated and trimmed random variables have different asymptotic behavior. Finally, we discuss resampling procedures which enable one to determine critical values in the case of small and moderate sample sizes.

Citation

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István Berkes. Lajos Horváth. Johannes Schauer. "Asymptotics of trimmed CUSUM statistics." Bernoulli 17 (4) 1344 - 1367, November 2011. https://doi.org/10.3150/10-BEJ318

Information

Published: November 2011
First available in Project Euclid: 4 November 2011

zbMATH: 1229.62017
MathSciNet: MR2854775
Digital Object Identifier: 10.3150/10-BEJ318

Keywords: change point , Resampling , Stable distributions , Trimming , weak convergence

Rights: Copyright © 2011 Bernoulli Society for Mathematical Statistics and Probability

Vol.17 • No. 4 • November 2011
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