Approximate and asymptotic distributions of chi-squared-type mixtures with applications

Abstract

In this article we study how to approximate a random variable T of general chi-squared-type mixtures by a random variable of the form aX d 2+ β matching the first three cumulants. The approximation error bounds for the density functions of the chi-squared approximation and the normal approximation are established. Applications of the results to some nonparametric goodness-of-fit tests, including those tests based on orthogonal series, smoothing splines, and local polynomial smoothers, are investigated. Two simulation studies are conducted to compare the chi-squared approximation and the normal approximation numerically. The chi-squared approximation is illustrated using a real data example for polynomial goodness-of-fit tests. © 2005 American Statistical Association.