The Decomposition of Quadratic Forms Under Skew Normal Settings

摘要

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.