A LILLIEFORS TEST OF FIT TO THE TWO-COMPONENT HOMOSCEDASTIC NORMAL MIXTURE

Abstract

The null distribution of a Lilliefors modification of the Kolmogorov-Smirnov test, was winvestigated as a goodness of fit test to the two-component homoscedastic normal mixture. The mixture parameters were estimated using a variable metric minimization algorithm to search for the maximum likelihood estimates. Selected percentiles of the null distribution were estimated from four sets of 2,500 pseudorandom samples, each of size 100, 150, 200, 250, 300, 350, 400, 450 and 500, with mixing proportions of p = 0.5, 0.6, 0.7, 0.8, 0.9, and 0.95, and standardized mean differences of d = 0.5, 1, 2, 3, 4, and 5 standard deviations apart (3,240,000 total samples).

The results suggest that the test is not invariant to the values of the mixture parameters, p and d. The critical values appeared to be well-behaved, strictly decreasing functions of the sample size. An essentially invariant test is developed by modeling the selected critical values as nonlinear functions of p, d, and n. Power studies suggest reasonable power for several alternatives.

Keywords:

• Empirical distribution function
• Kolmogorov-Smirnov test
• Likelihood ratio test

ACKNOWLEDGMENTS

Dr. Nancy Mendell's research on this paper was supported in part by NIH grant MH 49487.