Goodness-of-fit tests based on Kullback-Leibler discrimination information

We present a general methodology for developing asymptotically distribution-free goodness-of-fit tests based on the Kullback-Leibler discrimination information. The tests are shown to be omnibus within an extremely large class of nonparametric global alternatives and to have good local power. The proposed test procedure is a nonparametric extension of the classical Neyman-Pearson log-likelihood ratio test and is based on mth-order spacings between order statistics cross-validated by the observed log likelihood. The developed method also generalizes Cox´s procedure of testing separate families and covers virtually all parametric families of distributions encountered in statistics. It can also be viewed as a procedure based on sum-log functionals of nonparametric density-quantile estimators cross-validated by the log likelihood. With its good power properties, the method provides an extremely simple and potentially much better alternative to the classical empirical distribution function (EDF)-based test procedures. The important problem of selecting the order of spacings m in practice is also considered and a method based on maximizing the sample entropy constrained by the observed log likelihood is proposed. This data driven method of choosing m is demonstrated by Monte Carlo simulations to be more powerful than deterministic choices of m and thus provides a practically useful tool for implementing our test procedure