A new characterization of the Gamma distribution and associated goodness of fit tests

TL;DR

This paper introduces a new class of goodness-of-fit tests for the Gamma distribution based on a novel transformation and fixed point property, with proven theoretical properties and demonstrated effectiveness through simulations.

Contribution

It presents a new characterization of the Gamma distribution and develops associated goodness-of-fit tests with proven asymptotic properties and competitive performance.

Findings

Tests are consistent and have known weak limits.

Simulation shows competitive power against existing methods.

Method is based on a novel transformation linked to Stein's characterization.

Abstract

We propose a class of weighted $L_2$-type tests of fit to the Gamma distribution. Our novel procedure is based on a fixed point property of a new transformation connected to a Steinian characterization of the family of Gamma distributions. We derive the weak limits of the statistic under the null hypothesis and under contiguous alternatives. Further, we establish the global consistency of the tests and apply a parametric bootstrap technique in a Monte Carlo simulation study to show the competitiveness to existing procedures.