Distribution-Dependent Distance of First Two Moments

Closeness measures between distributions, between vectors, and between matrices abound. Many practical problems, however, call for measures of closeness between the first two moments of two unspecified distributions given only a sample of one of them or of a third distribution without other information. We present several metrics for such problems that demand special, distribution-dependent solutions, and show their good qualities. We demonstrate their rich applications in various areas, such as estimation performance analysis, efficiency of distributed fusion, metrized Kullback-Leibler divergence, decision performance evaluation, credibility of estimators, filter initialization, and empirical distribution function problems.

I. Introduction

Denote by a (vector-valued) random variable having mean and covariance regardless of its (specified or unspecified) distribution. Take estimation as an example. Then, can be an estimand, estimate, or estimation error, and is their corresponding first two moments.