The Decomposition of Quadratic Forms Under Skew Normal Settings

Abstract

In this paper, the decomposition properties of noncentral skew chi-square distribution is studied. A given random variable U having a noncentral skew chi-square distribution with \(k>1\) degrees of freedom, can be partitioned into the sum of two independent random variables \(U_1\) and \(U_2\) such that \(U_1\) has a noncentral skew chi-square distribution with 1 degree of freedom and \(U_2\) has the noncentral chi-square distribution with \(k-1\) degrees of freedom. Also if \(k>2\), this partition can be modified into \(U=U_1+U_2\), where \(U_1\) has a noncentral skew chi-square distribution with 2 degrees of freedom and \(U_2\) has a central chi-square distribution with \(k-2\) degrees of freedom. The densities of noncentral skew chi-square distributions with 1 degree of freedom, 2 degrees of freedom, and \(k>2\) degrees of freedom are derived, and their graphs are presented. For illustration of our main results, the linear regression model with skew normal errors is considered as an application.