Tests for location-scale families based on the empirical characteristic function

Abstract

A class of procedures is presented for using random samples to test the fit of location-scale families—distributions F(·;θ₁,θ₂) such that Z=(X−θ₁)/θ₂ has distribution Working with empirically standardized data, the test statistic is a measure of distance between the empirical characteristic function, and the c.f. of Z under the null hypothesis, ϕ₀(t). The closed-form test statistic is derived by integrating over the product of a weight function times Using as weight function for each location-scale family the squared modulus of ϕ₀ itself presents a unified test procedure. Included as special cases are well-known tests for normal and Cauchy laws. Small-sample powers are compared with those of Anderson-Darling tests for each of seven univariate location-scale families.