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The problem of selecting a model from two models with unknown location and scale parameters is considered. It is noted that the distribution of the ratio of maximum likelihoods does not depend upon the values of the nuisance location and scale parameters. Consequently, this ratio provides a convenient (and hopefully powerful) test for discriminating between two location and scale parameter models when the parameters are unknown. Tables for discriminating between the normal and the Cauchy; the normal and the exponential; or the normal and the double exponential are presented. An empirical comparison of the power of this test with the power of the UMP invariant test for discriminating between the normal and the Cauchy is given. Some further comparisons with the chi-square goodness of fit and the Kolmogorov-Smirnov tests for normality are also given.
Dumonceaux et al. (1973) studied this question.
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