Berry-Esseen Bounds for General Nonlinear Statistics, with Applications to Pearson's and Non-Central Student's and Hotelling's

• Activity Number: 135
• Type: Contributed
• Date/Time: Monday, July 30, 2012 : 8:30 AM to 10:20 AM
• Sponsor: IMS
• Author(s): Iosif Pinelis*+ and Raymond Molzon
• Companies: and Michigan Technological University
• Keywords: Berry-Esseen bounds ; non-central Student's statistic ; non-central Hotelling's statistic ; Pearson's correlation coefficient ; nonlinear statistics ; Stein method

Abstract

Uniform and nonuniform Berry--Esseen (BE) bounds of optimal orders on the closeness to normality for general nonlinear statistics are obtained. Applications to Student's, Pearson's, and Hotelling's statistics are given, which appear to be the first known results of these kinds (with the exception of uniform BE bounds for Student's statistic). The proofs use a Stein-type method developed by Chen and Shao, a Cramer-type of tilt transform, exponential and Rosenthal-type inequalities for sums of random vectors established by Pinelis, Sakhanenko, and Utev, as well as a number of other, quite recent results motivated by this study. The method allows one to obtain bounds with explicit and rather moderate-size constants, at least as far as the uniform bounds are concerned. For instance, one has a uniform BE bound for the Pearson sample correlation coefficient with the constant factor 4.08 (however, such a bound must necessarily involve the sixth moments of the marginals as well as the variance of the linear approximation to the Pearson statistic).