Distributions of Ratios of Two Correlated Skew-Normal Variables and of Ratios of Two Linear Functions of Order Statistics from Bivariate Normal Distribution

Abstract

In this article, we first derive the distribution of the ratio of two correlated skew-normal variables. Then, with , where |ρ|< 1, and (X _1:2, X _2:2) ^T denoting the corresponding order statistics, we consider two linear functions of these order statistics, viz., a ₁ X _1:2 + a ₂ X _2:2 and b ₁ X _1:2 + b ₂ X _2:2, and derive explicit expressions for the cumulative distribution function (cdf) and probability density function (pdf) of the ratios and .

Keywords:

• Bivariate normal distribution
• Bivariate skew-normal distribution
• Conditional Cauchy distribution
• Distribution of ratios
• Order statistics
• Orthant probabilities
• Skew-Cauchy distribution
• Skew-normal distribution
• Trivariate normal distribution

Mathematics Subject Classification:

• Primary 60E05
• Secondary 62H05

Acknowledgments

The first and last authors thank the Mahani Mathematical Research Center (Kerman, Iran) while the second author thanks the Natural Sciences and Engineering Research Council of Canada for supporting this research.