Tests of Coordinate Independence for a Bivariate Sample on a Torus

Abstract

This paper studies the problem of testing the independence of two random variables $X, Y$ from a random sample, $(X_1, Y_1), (X_2, Y_2) \cdots (X_n, Y_n)$, of size $n$, where $X$ and $Y$ are angular variates (i.e., reals modulo 1). In the standard case where $X$ and $Y$ are ordinary real variables, the following approach has been useful. Suppose $(X, Y)$ has the continuous distribution function $F(x, y)$ with marginal distribution functions $F_1(x)$ and $F_2(y)$, respectively. It is desired to test $H_0: F(x, y) = F_1(x)F_2(y)$ against the alternative $H_A: F(x, y) \neq F_1(x)F_2(y)$. Let $F_n(x, y)$ denote the sample distribution function of the random bivariate sample, i.e., if $H(x)$ denotes the left continuous Heaviside function then \begin{equation*}\tag{1}F_n(x, y) = \frac{1}{n} \sum^n_{j=1} H(x - X_j)H(y - Y_j).\end{equation*} Also, let $F_{n1}(x)$ and $F_{n2}(y)$ denote the sample distribution functions associated with the first and second components of the random sample vector. In terms of $H(x)$ \begin{equation*}\tag{2}F_{n1}(x) = \frac{1}{n} \sum^n_{j=1} H(x - X_j)\end{equation*} and \begin{equation*}\tag{3}F_{n2}(y) = \frac{1}{n} \sum^n_{j=1} H(y - Y_j).\end{equation*} Blum, Kiefer and Rosenblatt [1] studied the following distribution free tests of independence based on the sample distribution function. Reject for large values of \begin{equation*}\tag{4}A_n = \sup_{x,y}|T_n(x, y)|\end{equation*} or \begin{equation*}\tag{5}B_n = n \int\int\lbrack T_n(x, y)\rbrack^2 dF_n(x, y),\end{equation*} where \begin{equation*}\tag{6}T_n(x, y) = F_n(x, y) - F_{n1}(x)F_{n2}(y).\end{equation*} The first statistic, constructed in the spirit of Kolomorov-Smirnov statistics, has good power properties, cf. [1], Section 4, but its asymptotic distribution is unknown. The statistic $B_n$ is analogous to the Cramer-von Mises statistic and is also equivalent to a statistic originally proposed by Hoeffding [3]. The characteristic function of the null asymptotic distribution of $B_n$ is (cf. [1] and [3]) \begin{equation*}\tag{7}E e^{izB} = \prod^\infty_{j,k=1} \big(1 - \frac{2iz}{\pi^4j^2k^2}\big)^{-\frac{1}{2}}.\end{equation*} Asymptotic power properties are given in [1]. As in Rothman [9] the difficulty in modifying these tests in the toroidal case is that there is no natural origin for the distribution functions on a circle. Moreover, different arbitrary starting points give the test statistics $A_n$ and $B_n$ different values. In this paper we propose that the statistic $C_n$ be used for our problem, with the surface of the torus replacing the plane. \begin{equation*}\tag{8}C_n = n \int\int Z_n^2(x, y) dF_n(x, y)\end{equation*} where $Z_n(x, y) = T_n(x, y) + \int\int T_n(x, y) dF_n(x, y) - \int T_n(x, y) dF_{n1}(x) - \int T_n(x, y) dF_{n2}(y).$ We note that $C_n$ may be rewritten in the following form which we shall refer to in Section 3: \begin{equation*}\tag{9}C_n = 1/n^2 \sum^n_{j=1} \lbrack\sum^n_{a=1} \{T_n(X_j, Y_j) + T_n(X_a, Y_a) - T_n(X_a, Y_j) - T_n(X_j, Y_a)\}\rbrack^2.\end{equation*} We shall also have occasion to use the random variable, $D_n = n \int\int Z_n^{\ast^2}(x, y) dF_1(x)dF_2(y),$ where $Z_n^\ast(x, y) = T_n(x, y) + \int\int T_n(x, y)dF_1(x)dF_2(y) - \int T_n(x, y)dF_1(x) - \int T_n(x, y) dF_2(y).$ An outline of the paper follows: In Section 2 it is shown that when $H_0$ is true, $C_n - D_n \rightarrow 0$ in probability. The invariance of $C_n$ under changes of origin is proved in Section 3. Finally the asymptotic distribution of $C_n$ under the null hypothesis is obtained in Section 4.