關於某些無母數檢定統計量的漸近常態性之研究

Abstract

假定(X11,X21),…,(X1m,X2m)及(Y11,Y21),…,(Y1n,Y2n)是分別從具有連續分配函數Fx1,x2(x1,x2)及GY1,Y2 (y1,y2)之母體抽取出來的兩個獨立二元(bivariate)隨機樣本；我們假設這兩個母體具有相同之平均數(mean)，而這平均數可以是已知，也可以是未知。我們想要測出這兩個母體之離勢(variability or dispersion)是否不同。母體是一元(univariate)於情況已有甚多的研究發表。然而像這類母體是二元的情況，至今甚少研究。在這篇論文中，提出了兩種無母數檢定W及W*，並且在連續分配Fx1,x2(x1,x2)及GY1,Y2 (y1,y2)滿足相當一般的條件下，可得出W及W*的漸近常態性。這種漸近性在研究檢定過程的有效性(efficiency)是極為有用的。

Suppose (X11,X21),..., (X1m,X2m) and (Y11,Y21), ...,(Y1n,Y2n) are two independent bivariate random samples from populations with continuous distribution functions Fx1,x2 (x1,x2) and GY1,Y2 (y1,y2) respectively. We assume that the two populations have a common mean, which is either known or unknown. We would like to detect differences, in variability or dispersion for the two populations. However, the bivariate case seems to have been studied far less fully than a univariate one. In this paper, we suggest two nonparametric tests W and W* and establish the asymptotic normality of W and W* under fairly general conditions on the underlying distribution functions Fx1,x2 (x1,x2) and GY1,Y2 (y1,y2).This asymptotic property is very useful in investigating the efficiency of the test procedures.