# Bounding distributional errors via density ratios

## TL;DR

This paper introduces explicit bounds on distributional approximation errors using the maximal density ratio, providing a more informative measure than total variation distance, with applications to common distribution approximations.

## Contribution

It develops new explicit error bounds based on density ratios, applicable to various classical distribution approximation problems, with both upper and lower bounds.

## Key findings

- Provides explicit bounds for hypergeometric by binomial distributions
- Offers bounds for binomial by Poisson distributions
- Includes bounds for beta by gamma distributions

## Abstract

We present some new and explicit error bounds for the approximation of distributions. The approximation error is quantified by the maximal density ratio of the distribution $Q$ to be approximated and its proxy $P$. This non-symmetric measure is more informative than and implies bounds for the total variation distance. Explicit approximation problems include, among others, hypergeometric by binomial distributions, binomial by Poisson distributions, and beta by gamma distributions. In many cases we provide both upper and (matching) lower bounds.