Integral Transform Methods in Goodness-of-Fit Testing, I: The Gamma Distributions

TL;DR

This paper develops a new goodness-of-fit test for gamma distributions using Hankel transforms, deriving its distribution, properties, and efficiency under various alternatives, extending previous work on exponential distributions.

Contribution

It introduces a novel Hankel transform-based test for gamma distributions with detailed theoretical properties and efficiency analysis, expanding the scope beyond exponential models.

Findings

Derived the null distribution as an integrated squared Gaussian process.

Established the consistency and asymptotic properties of the test.

Analyzed the test's efficiency under local and contaminated alternatives.

Abstract

We apply the method of Hankel transforms to develop goodness-of-fit tests for gamma distributions with given shape parameter and unknown rate parameter, thereby extending results of Baringhaus and Taherizadeh (2010) on the exponential distributions. We derive the limiting null distribution of the test statistic as an integrated squared Gaussian process, obtain the corresponding covariance operator, and oscillation properties of its eigenfunctions. We show that the eigenvalues of the operator satisfy an interlacing property, and we apply that property in approximating critical values of the test statistic in one of the two applications to data considered. Further, we establish the consistency of the test. In studying properties of the test statistic under a variety of contiguous alternatives, we obtain the asymptotic distribution of the test statistic for gamma alternatives with varying rate or shape parameters and for a class of contaminated gamma models. We investigate the approximate Bahadur slope of the test statistic under local alternatives and we establish the validity of the Wieand condition, under which the approaches through the approximate Bahadur efficiency and the Pitman efficiency are in accord.