A Class of Nonparametric Tests for Independence in Bivariate Populations

Abstract

This paper is concerned with the problem of testing the independence of two random variables $X, Y$ on the basis of a random sample, $(X_1, Y_1), (X_2, Y_2), \cdots, (X_N, Y_N)$. The joint distribution function $H$ and, consequently, the marginal distribution functions $F$ and $G$ are assumed to be absolutely continuous. The hypothesis to be tested may be stated as $H(x, y) = F(x)G(y)$. The usual parametric test of this hypothesis is based on the sample correlation coefficient. Several nonparametric tests have been proposed and studied by Kendall [11], Hoeffding [8], [9], Blomqvist [1], and others. Konijn [13] has investigated the asymptotic power properties of some of these tests. The class of rank tests for independence to be considered in this paper is based on the test statistics of the form \begin{equation*}\tag{1.1}T_N = N^{-1}\sum^N_{i = 1} E_{N,r_i}E'_{N,s_i}Z'_{N,r_i}Z'_{N,s_i}\end{equation*} where $\{E_{N,i}\}, \{E'_{N,i}\}, i = 1, 2, \cdots, N$, are two sets of constants satisfying certain restrictions to be stated below, and $Z_{N,r_i} = 1 (Z'_{N,s_i} = 1)$ when $X_i(Y_i)$ is the $r_i$th ($s_i$th) smallest of the $X$'s ($Y$'s) and $Z_{N,r_i} = 0 (Z'_{N,s_i} = 0)$ otherwise. By taking $E_{N,r_i}(E'_{N,s_i})$ to be the expected value of the $r_i$th ($s_i$th) standard normal order statistic from a sample of size $N$, we get the normal scores test statistic which belongs to the class of $c_1$-statistics considered by Fisher and Yates [4], Hoeffding [10] and Terry [17]. If we put $E_{N,r_i} = r_i$ and $E_{N,s_i} = s_i$, the resulting test statistic is equivalent to the Spearman rank correlation statistic. The normal scores test is shown to be (a) the locally most powerful rank test and (b) asymptotically as efficient as the parametric correlation coefficient $\mathscr{R}$-test for the alternatives (4.1) and (4.2) when the underlying distributions are normal. It is at least as efficient as the $\mathscr{R}$-test when the alternative belongs to the class (4.1) and when $H$ satisfies the restrictions stated in Theorem 3. Tables 1 and 2 give the exact null distribution and some critical values of the normal scores statistic for $N \leqq 6$. Table 2 provides a comparison of the $t$-approximation with the exact distribution.