Optimal Goodness-of-Fit Tests for Location/Scale Families of Distributions Based on the Sum of Squares of $L$-Statistics

Abstract

A class of goodness-of-fit tests based on sums of squares of $L$-statistics is proposed for testing a composite parametric location and/or scale null hypothesis versus a general parametric alternative. It is shown that such tests can be constructed optimally to have the same asymptotic power against sequences of local alternatives as the generalized likelihood ratio statistics [G.L.R.S.] and, in fact, under suitable regularity conditions to be asymptotically equivalent to the G.L.R.S. One advantage of the proposed test statistic over the G.L.R.S. is that only an estimate of the scale parameter is needed in the computation of the statistic. No other parameter estimates are required. Also, an example of the practical implementation of the proposed hypothesis testing procedure is given.