## Bernoulli

• Bernoulli
• Volume 22, Number 4 (2016), 2029-2079.

### Cramér type moderate deviation theorems for self-normalized processes

#### Abstract

Cramér type moderate deviation theorems quantify the accuracy of the relative error of the normal approximation and provide theoretical justifications for many commonly used methods in statistics. In this paper, we develop a new randomized concentration inequality and establish a Cramér type moderate deviation theorem for general self-normalized processes which include many well-known Studentized nonlinear statistics. In particular, a sharp moderate deviation theorem under optimal moment conditions is established for Studentized $U$-statistics.

#### Article information

Source
Bernoulli, Volume 22, Number 4 (2016), 2029-2079.

Dates
Revised: August 2014
First available in Project Euclid: 3 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.bj/1462297674

Digital Object Identifier
doi:10.3150/15-BEJ719

Mathematical Reviews number (MathSciNet)
MR3498022

Zentralblatt MATH identifier
1272.68116

#### Citation

Shao, Qi-Man; Zhou, Wen-Xin. Cramér type moderate deviation theorems for self-normalized processes. Bernoulli 22 (2016), no. 4, 2029--2079. doi:10.3150/15-BEJ719. https://projecteuclid.org/euclid.bj/1462297674

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