# Goodness-of-fit Tests for the Bivariate Poisson Distribution

## TL;DR

This paper introduces a new goodness-of-fit test for the bivariate Poisson distribution, capable of detecting various alternatives, and reviews existing tests and extensions, supported by simulation results.

## Contribution

It proposes a novel, consistent goodness-of-fit test for the bivariate Poisson distribution that can detect local alternatives and employs bootstrap for null distribution estimation.

## Key findings

- Test is consistent against any fixed alternative.
- Bootstrap effectively estimates the null distribution.
- Simulation shows good power for finite samples.

## Abstract

The bivariate Poisson distribution is commonly used to model bivariate count data. In this paper we study a goodness-of-fit test for this distribution. We also provide a review of the existing tests for the bivariate Poisson distribution, and its multivariate extension. The proposed test is consistent against any fixed alternative. It is also able to detect local alternatives converging to the null at the rate $n^{-\frac{1}{2}}$. The bootstrap can be employed to consistently estimate the null distribution of the test statistic. Through a simulation study we investigated the goodness of the bootstrap approximation and the power for finite sample sizes.