Asymptotics of trimmed CUSUM statistics

TL;DR

This paper investigates the asymptotic behavior of trimmed CUSUM statistics in infinite variance settings, demonstrating their validity for change point detection after moderate trimming, and discusses resampling methods for small samples.

Contribution

It establishes the weak convergence of trimmed CUSUM processes to a Brownian bridge in infinite variance models, extending classical change point detection methods.

Findings

Trimmed CUSUM converges to a Brownian bridge in stable law domains

Classical change point tests remain valid after moderate trimming

Resampling procedures help determine critical values for small samples

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 $0<\alpha <2$, 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.