Brazilian Journal of Probability and Statistics

Change in the mean in the domain of attraction of the normal law via Darling–Erdős theorems

Miklós Csörgő and Zhishui Hu

Full-text: Open access

Abstract

This paper studies the problem of testing the null assumption of no-change in the mean of chronologically ordered independent observations on a random variable $X$ versus the at most one change in the mean alternative hypothesis. The approach taken is via a Darling–Erdős type self-normalized maximal deviation between sample means before and sample means after possible times of a change in the expected values of the observations of a random sample. Asymptotically, the thus formulated maximal deviations are shown to have a standard Gumbel distribution under the null assumption of no change in the mean. A first such result is proved under the condition that $EX^{2}\log\log(|X|+1)<\infty$, while in the case of a second one, $X$ is assumed to be in a specific class of the domain of attraction of the normal law, possibly with infinite variance.

Article information

Source
Braz. J. Probab. Stat., Volume 28, Number 4 (2014), 538-560.

Dates
First available in Project Euclid: 30 July 2014

Permanent link to this document
https://projecteuclid.org/euclid.bjps/1406741879

Digital Object Identifier
doi:10.1214/13-BJPS223

Mathematical Reviews number (MathSciNet)
MR3263064

Zentralblatt MATH identifier
1295.53077

Keywords
Change in the mean domain of attraction of the normal law Darling–Erdős theorems Gumbel distribution weighted metrics Brownian bridge

Citation

Csörgő, Miklós; Hu, Zhishui. Change in the mean in the domain of attraction of the normal law via Darling–Erdős theorems. Braz. J. Probab. Stat. 28 (2014), no. 4, 538--560. doi:10.1214/13-BJPS223. https://projecteuclid.org/euclid.bjps/1406741879


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