Smooth Goodness-of-Fit Tests: A Quantile Function Approach

Abstract

Abstract A quantile function based analog to Neyman's smooth tests for goodness-of-fit hypotheses is developed. These procedures possess asymptotic and small-sample power properties similar to those for smooth tests. They are, however, computationally simple, applicable to both complete and Type II censored data, and easily focused in the direction of any specific set of alternatives. These results are used to develop a new test for exponentiality. Only location/scale null hypotheses will be considered. The ideas are easily adapted to other types of nulls, however. Neyman first proposed smooth tests for the goodness-of-fit hypothesis. As stated by Kopecky and Pierce and by Rayner and Best, the basic idea behind these tests is to embed the null density in a larger exponential family. The procedure is then based upon the likelihood-ratio test statistic for testing that the additional parameters equal zero. Smooth type tests may be viewed as a compromise between omnibus test procedures, with generally low power in all directions, and procedures whose power is focused in the direction of a specific alternative. It is easily shown that any of the smooth type tests are inconsistent for some alternatives. Further, as stated by Koziol, one of the main criticisms of these tests has been that the larger exponential families employed seem to be selected more for mathematical convenience than as statistically reasonable alternative distributions. Results of Monte Carlo studies by Miller and Quesenberry and others, however, indicate that smooth tests perform well for a wide range of alternatives. The proposed tests are developed by embedding the quantile function associated with the null hypothesis into a larger linear model and then using typical regression techniques to test that the coefficients of the added terms in this model vanish. These statistics are interpreted and asymptotically have many of the properties of the corresponding regression tests. Further, as with Neyman's smooth tests, if the alternative model is properly selected, the resulting procedure provides a good omnibus test.