Empirical Orlicz norms

TL;DR

This paper investigates the properties of empirical Orlicz norms, establishing laws of large numbers and central limit theorems, and revealing unique convergence behaviors and limit distributions for these norms.

Contribution

It provides new theoretical results on the convergence and distributional properties of empirical Orlicz norms, including nonstandard rates and heavy-tailed limits.

Findings

Law of large numbers for empirical Orlicz norms

Central limit theorem with standard and nonstandard rates

Heavy-tailed stable limit distribution for certain cases

Abstract

The empirical Orlicz norm based on a random sample is defined as a natural estimator of the Orlicz norm of a univariate probability distribution. A law of large numbers is derived under minimal assumptions. The latter extends readily to a linear and a nonparametric regression model. Secondly, sufficient conditions for a central limit theorem with a standard rate of convergence are supplied. The conditions for the CLT exclude certain canonical examples, such as the empirical sub-Gaussian norm of normally distributed random variables. For the latter, we discover a nonstandard rate of $n^{1/4} \log(n)^{3/8}$, with a heavy-tailed, stable limit distribution. It is shown that in general, the empirical Orlicz norm does not admit any uniform rate of convergence for the class of distributions with bounded Orlicz norm.