Goodness-of-Fit Tests for Laplace, Gaussian and Exponential Power Distributions Based on $λ$-th Power Skewness and Kurtosis

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 $X^{\text{APD}}_{\lambda} := Z^2(S_{\lambda}) + Z^2(K^{\text{net}}_{\lambda})$ under the null and under local alternatives. We also develop an innovative regression strategy to obtain $Z$-scores, $Z(S_{\lambda})$ and $Z(K_{\lambda}^{\text{net}})$, that are nearly independent and distributed as standard Gaussians, resulting in a $\chi_2^2$ distribution valid for any sample of size (up to very high precision for $n\geq 20$). These two components can even be used as directional tests against asymmetric and symmetric alternatives, respectively. The case $\lambda=1$ leads to a powerful test of fit for the Laplace($\mu,\sigma$) distribution, whose empirical power is superior to all $38$ 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.