Heuristically Deciding Between Normal and Skew Normal Distributions for Describing the Data on a Response Variable and an Explanatory Variable

Abstract

Normal and skew normal distributions of a response variable Y for a given value x of an explanatory variable X are considered when the means of the distributions are linear functions of x. Deciding between these distributions for describing data is possible with the shape parameter of the skew normal distribution. The shape parameter can be either positive or negative. When the shape parameter is zero, a skew-normal distribution becomes a normal distribution. Larger magnitude of the shape parameter provides a better recognition of the distribution for describing the data. It is therefore important to estimate the shape parameter of the skew normal distribution along with the location and dispersion parameters. A linear approximation of the ratio of the standard normal density and distribution functions in the presence of the shape parameter of skew normal distribution is used for this purpose. A heuristic method is proposed to determine the sign and estimate the magnitude of shape parameter, and to estimate the location parameters: intercept and slope, and the dispersion parameter based on this linear approximation. Simulation studies for performance evaluation of the proposed heuristic method are presented.

Keywords:

• 62J05
• 62J86
• 62F10

Keywords:

• Likelihood
• Linear approximation
• Maximum likelihood
• Mill's ratio
• Model selection
• Normal distribution
• Shape parameter
• Skew normal distribution

Acknowledgment

The authors thank two reviewers for their careful reading of the earlier version of this article and their constructive suggestions.