Goodness-of-Fit Tests for the Skew-Normal Distribution When the Parameters Are Estimated from the Data

Abstract

In this article, tests are developed which can be used to investigate the goodness-of-fit of the skew-normal distribution in the context most relevant to the data analyst, namely that in which the parameter values are unknown and are estimated from the data. We consider five test statistics chosen from the broad Cramér–von Mises and Kolmogorov–Smirnov families, based on measures of disparity between the distribution function of a fitted skew-normal population and the empirical distribution function. The sampling distributions of the proposed test statistics are approximated using Monte Carlo techniques and summarized in easy to use tabular form. We also present results obtained from simulation studies designed to explore the true size of the tests and their power against various asymmetric alternative distributions.

Keywords:

• Asymmetry
• Cramér–von Mises
• Empirical distribution function
• Kolmogorov–Smirnov

Mathematics Subject Classification:

• Primary 62G10
• Secondary 62G30

Acknowledgments

This research has been supported by the Dirección General de Enseñanza Superior (DGES) of the Spanish Ministry for Education and Culture through the project BFM2003-05640/MATE.