Shapiro–Wilk test for skew normal distributions based on data transformations

ABSTRACT

A probability property that connects the skew normal (SN) distribution with the normal distribution is used for proposing a goodness-of-fit test for the composite null hypothesis that a random sample follows an SN distribution with unknown parameters. The random sample is transformed to approximately normal random variables, and then the Shapiro–Wilk test is used for testing normality. The implementation of this test does not require neither parametric bootstrap nor the use of tables for different values of the slant parameter. An additional test for the same problem, based on a property that relates the gamma and SN distributions, is also introduced. The results of a power study conducted by the Monte Carlo simulation show some good properties of the proposed tests in comparison to existing tests for the same problem.

KEYWORDS:

• Goodness of fit
• tests of hypotheses
• normality test
• Anderson–Darling test
• skew data sets
• Monte Carlo simulation
• parametric bootstrap

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 62G10
• 62F40
• 62E10

Acknowledgments

The authors are grateful to the anonymous reviewers and associate editor for their valuable comments and suggestions on the original version of this paper, which helped to improve its current presentation. The authors are also grateful to Arturo Mancera-Rico for allowing us to include the strain data set used in Section 5 and to J. A. Villaseñor for useful discussions.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Elizabeth González-Estrada  http://orcid.org/0000-0002-3086-0605

Waldenia Cosmes  http://orcid.org/0000-0003-4271-694X