Goodness-of-fit tests for Laplace, Gaussian and exponential power distributions based on $\lambda$-th power skewness and kurtosis

TL;DR

This paper develops new goodness-of-fit tests for Laplace, Gaussian, and exponential power distributions using extended skewness and kurtosis measures, with superior empirical power demonstrated on temperature data.

Contribution

It introduces a novel family of tests based on Rao's score for the exponential power distribution, including innovative regression methods for Z-score independence, and provides extensive theoretical and empirical validation.

Findings

The tests outperform 39 competitors in power across diverse alternatives.

The asymptotic distribution of the test statistic is derived under null and alternative hypotheses.

The methods are successfully applied to temperature datasets, showing practical utility.

Abstract

Temperature data, like many other measurements in quantitative fields, are usually modeled using a normal distribution. However, some distributions can offer a better fit while avoiding underestimation of tail event probabilities. To this point, we extend Pearson's notions of skewness and kurtosis to build a powerful family of goodness-of-fit tests based on Rao's score for the exponential power distribution $\mathrm{EPD}_{\lambda}(\mu,\sigma)$, including tests for normality and Laplacity when $\lambda$ is set to 1 or 2. We find the asymptotic distribution of our test statistic, which is the sum of the squares of two $Z$-scores, under the null and under local alternatives. We also develop an innovative regression strategy to obtain $Z$-scores that are nearly independent and distributed as standard Gaussians, resulting in a $\chi_2^2$ distribution valid for any sample size (up to very high precision for $n\geq 20$). The case $\lambda=1$ leads to a powerful test of fit for the Laplace($\mu,\sigma$) distribution, whose empirical power is superior to all $39$ competitors in the literature, over a wide range of $400$ alternatives. Theoretical proofs in this case are particularly challenging and substantial. We applied our tests to three temperature datasets. The new tests are implemented in the R package PoweR.