Bounds on Variance for Unimodal Distributions

TL;DR

This paper establishes bounds on the variance of unimodal distributions using differential entropy, showing how entropy can serve as a surrogate for distribution concentration within specific subclasses.

Contribution

It provides new upper and lower bounds on variance in terms of entropy power for subclasses of unimodal distributions, linking entropy and variance more tightly.

Findings

Variance is bounded above and below by scaled entropy power.

Differential entropy decreases exponentially with variance.

Entropy can be used as a surrogate for distribution concentration.

Abstract

We show a direct relationship between the variance and the differential entropy for subclasses of symmetric and asymmetric unimodal distributions by providing an upper bound on variance in terms of entropy power. Combining this bound with the well-known entropy power lower bound on variance, we prove that the variance of the appropriate subclasses of unimodal distributions can be bounded below and above by the scaled entropy power. As differential entropy decreases, the variance is sandwiched between two exponentially decreasing functions in the differential entropy. This establishes that for the subclasses of unimodal distributions, the differential entropy can be used as a surrogate for concentration of the distribution.