Boundary kernel estimation of the two sample comparison density function

Abstract

The focus of this work is to derive functional and graphical statistical techniques for the two sample problem suitable for implementation in modern computing environments. In the two sample problem, it is desired to test the null hypothesis that two independent random samples have a common distribution function. Assuming certain conditions on the distribution functions, a procedure is proposed which has strong graphical elements, a sound theoretical foundation, and estimates the relation of the two distributions if the null hypothesis is rejected. The proposed procedure has as its motivation the estimation of the comparison density and inference concerning its uniformity. The proposed procedure is both a statistical test of the null hypothesis and a model selection criterion. The test is based on components of a new stochastic process which is termed the kernel density process. This process is based on a boundary kernel estimate of the comparison density. It is proposed to apply a new test, the subset chi-square test, to these components. If the null hypothesis is rejected, the components found to be significant are used to construct a damped orthogonal series estimate of the comparison density. The power of the proposed test under local alternatives is compared to two commonly used portmanteau statistics, the Cramer-von Mises and the Anderson-Darling, and to a third statistic suggested by this work. A new method for finding the power of these statistics under local alternatives is given. This method uses the fast Fourier transform to invert an approximation to the characteristic function of the statistic. The proposed test is seen to have good power properties. A simulation study is conducted to examine its small sample size. Its size is found to remain close to its nominal value.