The Annals of Statistics

Self-normalized Cramér-type moderate deviations under dependence

Xiaohong Chen, Qi-Man Shao, Wei Biao Wu, and Lihu Xu

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Abstract

We establish a Cramér-type moderate deviation result for self-normalized sums of weakly dependent random variables, where the moment requirement is much weaker than the non-self-normalized counterpart. The range of the moderate deviation is shown to depend on the moment condition and the degree of dependence of the underlying processes. We consider three types of self-normalization: the equal-block scheme, the big-block-small-block scheme and the interlacing scheme. Simulation study shows that the latter can have a better finite-sample performance. Our result is applied to multiple testing and construction of simultaneous confidence intervals for ultra-high dimensional time series mean vectors.

Article information

Source
Ann. Statist. Volume 44, Number 4 (2016), 1593-1617.

Dates
Received: April 2015
Revised: December 2015
First available in Project Euclid: 7 July 2016

Permanent link to this document
http://projecteuclid.org/euclid.aos/1467894709

Digital Object Identifier
doi:10.1214/15-AOS1429

Mathematical Reviews number (MathSciNet)
MR3519934

Subjects
Primary: 62E20: Asymptotic distribution theory 60F10: Large deviations

Keywords
Cramér-type moderate deviation absolutely regular functional dependence measures ultra-high dimensional time series

Citation

Chen, Xiaohong; Shao, Qi-Man; Wu, Wei Biao; Xu, Lihu. Self-normalized Cramér-type moderate deviations under dependence. Ann. Statist. 44 (2016), no. 4, 1593--1617. doi:10.1214/15-AOS1429. http://projecteuclid.org/euclid.aos/1467894709.


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Supplemental materials

  • Supplement to “Self-normalized Cramér type moderate deviations under dependence”. The supplement gives the detailed proof for Theorem 4.2.