Tests of Distributional Hypotheses with Nuisance Parameters Using Fourier Series Methods
基于傅里叶系数开发了一类平滑卡方检验，用于检验分布假设，包括完全指定分布和含未知参数的复合零假设，并通过模拟验证了其收敛速度和检验功效。
We develop a class of smoothed χ 2 tests of distributional hypotheses based on Fourier coefficients used in the density estimation procedure of Fellner. Starting with the simple test of Fx = G, where G is a completely specified hypothesized continuous distribution, we develop tests of the composite null hypothesis Fx − Gα for some vector of unknown parameters α. Variances used in computing the tests vary according to the choice of distributional family and depend on estimators used to estimate the nuisance parameters α. 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 χ 2 distribution under the null hypothesis. The power of the test of normality was found to be competitive with other, more specialized tests.