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

Abstract

Temperature data, like many other measurements in quantitative fields, are usually modelled 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 λ 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 $n\ge 20$). 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.

Keywords:

• asymmetric power distribution
• Lagrange multiplier test
• local alternatives
• power analysis
• Rao's score test
• temperature data

2020 MSC:

• Primary: 62F03
• Secondary: 62E20
• 62F12
• 60F05

Acknowledgments

We thank the anonymous referee for his/her comments. This research includes computations performed using the computational cluster Katana supported by Research Technology Services at UNSW Sydney.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Source: Q62 in XLS file at https://www.foodrisk.org/resources/sendFile/49 and see https://www.foodrisk.org/resources/sendFile/46 for a description of the format.

2 Source: https://www.fda.gov/consumers/consumer-updates/are-you-storing-food-safely

Additional information

Funding

F. Ouimet was supported by postdoctoral fellowships from the Natural Sciences and Engineering Research Council of Canada (PDF) and the Fond Québécois de la Recherche sur la Nature et les Technologies (B3X supplement and B3XR). F. Ouimet is currently supported by a CRM-Simons postdoctoral fellowship from the Centre de recherches mathématiques and the Simons foundation.