On new tests for the Poisson distribution based on empirical weight functions

TL;DR

This paper introduces new goodness-of-fit tests for the Poisson distribution using empirical weight functions and weighted Lp distances, demonstrating improved power over existing tests, especially for overdispersed data.

Contribution

The paper develops novel weighted Lp distance-based tests for the Poisson distribution, including closed-form expressions and practical applications.

Findings

New tests are powerful against overdispersed alternatives

Closed-form formulas for L1, L2, and L1 distances are derived

Monte Carlo simulations show improved performance over existing tests

Abstract

We propose new goodness-of-fit tests for the Poisson distribution. The testing procedure entails fitting a weighted Poisson distribution, which has the Poisson as a special case, to observed data. Based on sample data, we calculate an empirical weight function which is compared to its theoretical counterpart under the Poisson assumption. Weighted Lp distances between these empirical and theoretical functions are proposed as test statistics and closed form expressions are derived for L1, L2 and L1 distances. A Monte Carlo study is included in which the newly proposed tests are shown to be powerful when compared to existing tests, especially in the case of overdispersed alternatives. We demonstrate the use of the tests with two practical examples.