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 via 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.