## Bernoulli

• Bernoulli
• Volume 17, Number 4 (2011), 1344-1367.

### Asymptotics of trimmed CUSUM statistics

#### 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.

#### Article information

Source
Bernoulli, Volume 17, Number 4 (2011), 1344-1367.

Dates
First available in Project Euclid: 4 November 2011

https://projecteuclid.org/euclid.bj/1320417507

Digital Object Identifier
doi:10.3150/10-BEJ318

Mathematical Reviews number (MathSciNet)
MR2854775

Zentralblatt MATH identifier
1229.62017

#### Citation

Berkes, István; Horváth, Lajos; Schauer, Johannes. Asymptotics of trimmed CUSUM statistics. Bernoulli 17 (2011), no. 4, 1344--1367. doi:10.3150/10-BEJ318. https://projecteuclid.org/euclid.bj/1320417507

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