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Empirical Orlicz norms

Fabian Mies

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.

Empirical Orlicz norms

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 , 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.

Paper Structure

This paper contains 3 sections, 10 theorems, 52 equations.

Table of Contents

  1. Motivation
  2. Law of large numbers
  3. Central limit theorem

Key Result

Theorem 2.1

For $X_1,\ldots, X_n\overset{iid}{\sim} X$ such that $\|X\|_{\psi}<\infty$,

Theorems & Definitions (19)

  • Theorem 2.1: Law of large numbers
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 3.1: Central limit theorem
  • Example : Exponential distribution
  • Proposition 3.2
  • Example : Weibull distribution
  • Proposition 3.3
  • Example : Normal distribution
  • Proposition 3.4
  • ...and 9 more