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

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. The result on the limit null distribution is used to prove the asymptotic validity of the parametric bootstrap that is implemented to run the tests. Further, we establish the global consistency of our tests in this bootstrap setting, and conduct a Monte Carlo simulation study to show the competitiveness to existing test procedures.