The noncentral chi-squared distribution with zero degrees of freedom and testing for uniformity

Abstract

The noncentral chi-squared distribution with zero degrees of freedom is defined to be a Poisson mixture of mass at zero together with chi-squared distributions that have even degrees of freedom. The name is justified by the decomposition of the classical noncentral chi-squared distribution as the sum of a central chi-squared component having the full number of degrees of freedom and an independent noncentral chi-squared component having zero degrees of freedom. The basic properties of this one parameter family of distributions are given, and they are shown to be useful in the computation of approximate critical values of a test for uniformity.