Sharp Local Minimax Rates for Goodness-of-Fit Testing in multivariate Binomial and Poisson families and in multinomials

TL;DR

This paper establishes sharp local minimax rates for goodness-of-fit testing in multivariate binomial, Poisson, and multinomial distributions, providing simple, implementable tests that adapt to the known distribution.

Contribution

It introduces locally minimax-optimal goodness-of-fit tests for multivariate discrete distributions with explicit thresholds and simple forms, advancing the understanding of detection boundaries.

Findings

Characterized the detection threshold for various distances $d(p,q)$.

Developed simple, implementable tests with optimal local minimax properties.

Provided theoretical analysis for multivariate binomial, Poisson, and multinomial families.

Abstract

We consider the identity testing problem - or goodness-of-fit testing problem - in multivariate binomial families, multivariate Poisson families and multinomial distributions. Given a known distribution $p$ and $n$ iid samples drawn from an unknown distribution $q$, we investigate how large $\rho>0$ should be to distinguish, with high probability, the case $p=q$ from the case $d(p,q) \geq \rho $, where $d$ denotes a specific distance over probability distributions. We answer this question in the case of a family of different distances: $d(p,q) = \|p-q\|_t$ for $t \in [1,2]$ where $\|\cdot\|_t$ is the entrywise $\ell_t$ norm. Besides being locally minimax-optimal - i.e. characterizing the detection threshold in dependence of the known matrix $p$ - our tests have simple expressions and are easily implementable.