TESTS OF DISTRIBUTIONAL HYPOTHESES WITH NUISANCE PARAMETERS USING FOURIER-SERIES METHODS

We develop a class of smoothed chi-2 tests of distributional hypotheses based on Fourier coefficients used in the density estimation procedure of Fellner. Starting with the simple test of F(x) = G, where G is a completely specified hypothesized continuous distribution, we develop tests of the composite null hypothesis F(x) = G-alpha for some vector of unknown parameters alpha. Variances used in computing the tests vary according to the choice of distributional family and depend on estimators used to estimate the nuisance parameters alpha. Expressions are given that allow one to use fast Fourier transforms to compute the necessary quantities. Examples given include tests of exponential, normal, Laplace, Pareto, Weibull, and uniform distributions as well as a test of the normality of residuals in least squares regression. Simulation studies indicate fast convergence of the statistics' distributions to the chi-2 distribution under the null hypothesis. The power of the test of normality was found to be competitive with other, more specialized tests.