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.

Published at http://dx.doi.org/10.3150/15-BEJ719 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

References in corpus (2)

• Normal approximation for nonlinear statistics using a concentration inequality approach
• Necessary and sufficient conditions for the asymptotic distributions of coherence of ultra-high dimensional random matrices

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• Estimating the Penalty Level of ${\ell }_1$-minimization via Two Gaussian Approximation Methods
• Self-normalized Cramér-type Moderate Deviation of Stochastic Gradient Langevin Dynamics
• Two-Sample Smooth Tests for the Equality of Distributions
• Self-Normalized Moderate Deviations for Degenerate U-Statistics
• Berry--Esseen bounds for self-normalized sums of local dependent random variables