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E-variables and tests of randomness for distribution classes

Georgii Potapov, Yuri Kalnishkan

TL;DR

This paper introduces the method of e-variable-approximability and uses it to develop a general approximation technique allowing us to construct e-variables for popular distribution classes important for applications.

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.

E-variables and tests of randomness for distribution classes

TL;DR

This paper introduces the method of e-variable-approximability and uses it to develop a general approximation technique allowing us to construct e-variables for popular distribution classes important for applications.

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.
Paper Structure (20 sections, 14 theorems, 55 equations)

This paper contains 20 sections, 14 theorems, 55 equations.

Table of Contents

  1. Introductions
  2. History
  3. Our contribution
  4. Preliminaries
  5. E-variables
  6. Computable and semicomputable functions
  7. Randomness tests
  8. Organisation
  9. $E$-variable-approximability
  10. Main results on $e$-variable-approximability
  11. Interpolating approximation
  12. Lower semicomputable approximations of randomness deficiency $\inf$-projection
  13. Discussions
  14. Discussion of definition \ref{['dfn:e-val-approx']}
  15. Further research
  16. ...and 5 more sections

Key Result

Theorem 1

For $\mathcal{X}$ that is of the form $\{0,1\}^*, \{0,1\}^\omega, \mathbb{R}^n$ and $\mathcal{Y}$ that is either one of these sets or the set of bounded measures on such sets, there exists a uniform randomness test $\mathbf{t}$ such that for any uniform randomness test $f$ there is a constant $C_f > holds for any $x, \mathbb{P}, y$. Any such test is called a universal uniform randomness test, or s

Theorems & Definitions (40)

  • Definition 2.1: $p$-variable
  • Definition 2.2: $e$-variable
  • Definition 2.3
  • Definition 2.4
  • Remark
  • Definition 2.5: Randomness test
  • Theorem 1: existence and definition of a universal uniform test (levin1976uniformGACS200591VovkVyuginBayes)
  • Definition 2.6
  • Theorem 2: VovkLearningBernoulli199796vovk2016concept
  • Definition 3.1
  • ...and 30 more