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Varentropy Estimation via Nearest Neighbor Graphs

Nikolai Leonenko, Yu Sun, Emanuele Taufer

Abstract

The Varentropy is a measure of the variability of the information content of random vector and it is invariant under affine transformations. We introduce the statistical estimate of varentropy of random vector based on the nearest neighbor graphs (distances). The asymptotic unbiasedness and L2-consistency of the estimates are established.

Varentropy Estimation via Nearest Neighbor Graphs

Abstract

The Varentropy is a measure of the variability of the information content of random vector and it is invariant under affine transformations. We introduce the statistical estimate of varentropy of random vector based on the nearest neighbor graphs (distances). The asymptotic unbiasedness and L2-consistency of the estimates are established.
Paper Structure (26 sections, 12 theorems, 260 equations)

This paper contains 26 sections, 12 theorems, 260 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Proof of Theorem 1
  4. Proof of Lemma \ref{['result I.1']}
  5. Proof of Theorem \ref{['result II']}
  6. A Stronger Statement
  7. Proof of $\mathsf{Var}(S^{2}_{N}) \,\xrightarrow[]{}\, 0$
  8. Boundedness of $\mathsf{Var}[ \tilde{\zeta}_{1}^{2}(N)]$
  9. Finiteness of $I_{1}(N,x)$
  10. Finiteness of $I_{2}(N,x)$
  11. Finiteness of $J_{1}(N,x)$
  12. Finiteness of $J_{2}(N,x)$
  13. Investigation of $J_{2.1}(N,x)$
  14. Investigation of $J_{2.2}(N,x)$
  15. Proof of $\mathsf{Cov}[ \tilde{\zeta}_{i}^{2}(N)\,,\,\tilde{\zeta}_{j}^{2}(N)]\xrightarrow[]{N\rightarrow \infty}\,0$
  16. ...and 11 more sections

Key Result

Theorem 1

Assume that, for some positive $\varepsilon_{i},\,R_{j}$, where $i= 0,1,2$ and $j=1,2$, the functionals appearing of Eqs.(condition K), (condition Q), (condition T) are finite for ${\color{black} \alpha=1,2}$, that is $K_{f,\,{\color{black} \alpha}}(\varepsilon_{0})<\infty,\,{\color{black} \alpha=1,

Theorems & Definitions (22)

  • Theorem 1
  • Theorem 2
  • Corollary 1
  • Corollary 2
  • Corollary 3
  • Lemma 1
  • Lemma 2
  • proof : Proof of Lemma \ref{['lemma power 3']}
  • Lemma 3
  • proof : Proof of Lemma \ref{['lemma power 4']}
  • ...and 12 more