Test Statistics Derived as Components of Pearson's Phi-Squared Distance Measure
本文提出基于皮尔逊θ²度量的分量方法，通过比较密度和傅里叶分析统一推导多种经典检验统计量（如Wilcoxon、t统计量、F检验等），并生成新检验程序，适用于单样本、两样本、多样本及独立性检验问题。
Abstract A general approach to the comparison of two absolutely continuous distributions is presented based on Pearson's θ2 measure. Although no simple estimates are available for θ2, it can be decomposed into components, all of which are easily estimated. Asymptotic distribution theory is derived for the component estimates, and the form of the components is considered for several standard problems. It is demonstrated that the components-of-θ2 approach provides a unified approach to the development of tests for a variety of problems and that it is useful for generating new test procedures. The components-of-θ2 approach is based on what is termed a comparison density. That is, given two absolutely continuous distributions F and H with densities f and h, the comparison density is defined as d(u) = f(H –1(u))/h(H –1(u)), where H –1 is the H quantile function. Since d must be uniform under the hypothesis F = H, an overall measure of the validity of the hypothesis is provided by φ2 = ∫1 0 (d(u) − 1)2 du. By using Fourier analysis, θ2 can be written as Σ∞ j=1 = a2 j, where the aj 's are Fourier coefficients of d under an orthonormal basis for the set of square-integrable functions. Since H = F iff all of the aj 's vanish, the aj 's correspond to subhypotheses about the disparity between F and H. Test statistics for these subhypotheses can be based on estimates of the aj 's that are easy to construct in many cases. The estimates of the components aj for certain problems are found to have some familiar forms. In the case of one-sample tests of symmetry, the Wilcoxon and normal-scores test statistics can be derived as specific components of θ2. For the one-sample goodness-of-fit problem one obtains statistics such as the t statistic, the usual chi-squared statistic for testing hypotheses about the variance, the classical third-moment statistic for tests of symmetry, and the Neyman "smooth" statistics. When samples have been obtained from k ≥ 2 distributions F 1, …, Fk to test F 1 =… = Fk , the estimation of appropriate Fourier coefficients yields the Wilcoxon, Mood, normal-scores, and Klotz tests for k = 2 and their parallels for k > 2. The classical F test for one-way analysis of variance can also be obtained from this viewpoint. For bivariate tests of independence the components-of-θ2 approach produces test statistics that include Spearman's ρ. Along with these familiar statistics a variety of new test statistics of potential value can also be derived. Key Words: Comparison densityMultisample problemsOne-sample problemsTwo-sample problemsRank testsRegressionTests for independence