Exact Kolmogorov and total variation distances between some familiar discrete distributions

José A Adell,P Jodrá

doi:10.1155/jia/2006/64307

Abstract

We give exact closed-form expressions for the Kolmogorov and the total variation distances between Poisson, binomial, and negative binomial distributions with different parameters. In the Poisson case, such expressions are related with the Lambert function.

Highlights

Estimates of the closeness between probability distributions measured in terms of certain distances, the Kolmogorov and the total variation distances are very common in theoretical and applied probability
As far as we know, only a few exceptions deal with exact formulae
Denote by Sn(p) a random variable having the binomial distribution with parameters n and p, that is, P Sn(p) = k :=

Summary

Introduction

Estimates of the closeness between probability distributions measured in terms of certain distances, the Kolmogorov and the total variation distances are very common in theoretical and applied probability. As far as we know, only a few exceptions deal with exact formulae (see, e.g., Kennedy and Quine [5], where the exact total variation distance between binomial and Poisson distributions is given for small values of the success parameter of the binomial). The aim of this note is to provide exact closed-form expressions for the Kolmogorov and the total variation distances between Poisson, binomial, and negative binomial distributions with different parameters.

Results

Conclusion