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

摘要

In this article, we first derive the distribution of the ratio of two correlated skew-normal variables. Then, with (X_1, X_2)~T ～ N_2(0, 0, σ_1~2, σ_2~2, ρ), 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_1X_(1:2) + a_2X_(2:2) and b_1X_(1:2) + b_2X_(2:2), and derive explicit expressions for the cumulative distribution function (cdf) and probability density function (pdf) of the ratios Y_1 = (a_1 X_(1:2)+a_2 X_(2:2))/(b_1 X_(1:2)+b_2 X_(2:2)) and Y_2 = (a_1 X_(1:2)+a_2 X_(2:2))/(|b_1 X_(1:2)+b_2 X_(2:2)|).