E-variables and tests of randomness for distribution classes

TL;DR

This paper introduces a new approximation method for constructing e-variables, enabling effective randomness tests for various distribution classes, advancing statistical hypothesis testing techniques.

Contribution

It develops the e-variable-approximability method, allowing explicit construction of e-variables for important distribution classes in statistical testing.

Findings

Provides a general approximation technique for e-variables.

Enables explicit construction of randomness tests for distribution classes.

Advances the application of e-variables in hypothesis testing.

Abstract

E-variables are a relatively new approach for testing statistical hypotheses that has been experiencing major development during the last several years. In this paper we introduce the method of e-variable-approximability and use it to develop a general approximation technique allowing us to construct e-variables for popular distribution classes important for applications. E-variables were originally based on a concept of Levin's (average-bounded) randomness tests from Algorithmic Information Theory. We show that our construction of e-variables can be used to provide an explicit construction for a randomness test with respect to a class of distributions.