A Layer Rank Test for Ordered Bivariate Alternatives

Abstract

Let $Z_i = (X_i, Y_i), i = 1,2,\cdots, m$ be a random sample from a bivariate population with continuous $\operatorname{cdf} F(x, y)$ and let $Z_i = (X_i, Y_i), i = m + 1, \cdots, m + n$ be an independent sample from the population with continuous $\operatorname{cdf} G(x, y)$. The present paper is concerned with the development of a nonparametric procedure for testing the null hypothesis $H:F = G$ against a class of one-sided alternatives which are, generally, those $\operatorname{cdf's}$ which are shifted from $F$ towards large values of $x$ and $y$. That is, large values of $x$ and $y$ are more likely under $G$ than $F$. A formal definition of the alternatives, tied to a concept of two-dimensional ordering, is introduced in Section 2. A simple example is given by the one-sided translation alternatives \begin{equation*}\tag{1.1}K_0 = \{(F, G): G(x, y) = F(x - \theta_1, y - \theta_2), \mathbf{\theta} \geqq \mathbf{0} \text{and} \mathbf{\theta} \neq \mathbf{0}\}\end{equation*} where inequality between vectors means coordinate-wise inequality. Another interesting class of directed shifts, which has not been previously studied, is composed of the bivariate Lehmann-type alternatives \begin{equation*}\tag{1.2}K_1 = \{(F, G):G(x, y) = F^\theta(x, y) \theta > 1\}.\end{equation*} All $\operatorname{cdf's}$ are assumed to be continuous unless otherwise stated. Among other areas, problems involving ordered alternatives arise in reliability studies of two-component systems when the components are interdependent and prior information on the mechanism involved ensures that the average lifetimes of the components in one system are no smaller than those of the other. Another example would be the comparison of a new with a standard method for treating a disease when two responses are measured. The usefulness of incorporating prior information in the form of an ordered parameter space was first recognized by Bartholomew [2], [3], [4] in the one-way analysis of variance model. He derived the likelihood ratio test for the hypothesis that the means of several normal distributions are equal against the alternative that they are ordered. Maximum likelihood estimation of ordered parameters in other distributions has been studied by Brunk [5] and Van Eeden [17]. Chacko [6] and Shorack [16] proposed rank analogs of Bartholomew's test and showed that the asymptotic Pitman efficiency equals that of the Wilcoxon test relative to the $t$ test. Employing some results from nonlinear programming theory, Kudo [9] and Nuesch [12] derived the likelihood ratio procedure for testing $\mathbf{\mu} = \mathbf{0}$ against $\mathbf{\mu} \geqq \mathbf{0}$ on the basis of a single sample from a multivariate normal $\mathcal{N}(\mathbf{\mu}, \mathbf{\Sigma})$ with $\mathbf{\Sigma}$ known. The case of unknown $\mathbf{\Sigma}$ has recently been investigated by Perlman [13]. Other tests when $\not\sum$ is known, namely most stringent somewhere most powerful tests, have been studied by Schaafsma and Smid [15] and these have been compared with most stringent tests by Schaafsma [14]. No distribution-free or even asymptotically distribution-free test is yet available in the literature for detecting ordered shifts in bivariate and multivariate distributions. As groundwork for the bivariate two-sample problem, we propose and study a nonparametric test based on the concept of two-dimensional layer ranks which were introduced by Barndorff-Nielsen and Sobel [1]. Let $N = m + n$ and define \begin{align*}L(i, j) &= 1 \quad\text{if}\mathbf{Z}_i \geqq \mathbf{Z}_j; \\ &= 0\quad \text{otherwise}.\end{align*}\begin{equation*}\tag{1.3}\begin{align*}L_i &= \sum^N_{j = 1} L(i, j),\quad L_j^\ast = \sum^N_{i = 1} L(i, j);\quad 1 \leqq i \leqq N, 1 \leqq j \leqq N, \\ \mathbf{L} &= (L_1, \cdots, L_N),\quad \mathbf{L}^\ast = (L_1^\ast, \cdots, L_N^\ast).\end{align*}\end{equation*} Then $L_i(L_i^\ast)$ is called the 3rd (1st) quadrant layer rank of $\mathbf{Z}_i$ in the combined sample $\{\mathbf{Z}_1, \cdots, \mathbf{Z}_N\}$. Geometrically, $L_i$ and $L_i^\ast$ are the number of points $\mathbf{Z}_j$ in the closed 3rd and 1st quadrant, respectively, with reference to Cartesian coordinates having origin $\mathbf{Z}_i$. For intuitive motivation of our test statistic, consider plotting the first sample as dots and the second as crosses in the same diagram [Fig. 1]. Under $H$, the $\mathbf{Z}_i$ are independent identically distributed and the dots and crosses are expected to be well mixed. Under an ordered alternative, $G$ has more mass in the upper right-hand corner than $F$ and the second sample layer ranks $L_{m + 1}, \cdots, L_N$ are expected to be larger than $L_1, \cdots, L_m$ on the average. High values of the statistic $\lbrack m\sum^N_{i = m + 1} L_i - n \sum^m_{i = 1} L_i\rbrack$ should then lead to rejection of $H$ in favor of an ordered shift. Similarly, a small value of $\lbrack m \sum^N_{i = m + 1} L_i^\ast - n \sum^m_{i = 1} L_i^\ast\rbrack$ indicates a one-sided shift. Incidentally, for the univariate two sample problem the diagram would be one of dots and crosses on a line and the layer ranks would reduce to ordinary ranks making each of the two statistics equivalent to the one-sided Wilcoxon statistic. Although $\mathbf{L}$ and $\mathbf{L}^\ast$ are invariant under a natural group of transformations which leave the problem invariant, it is difficult to characterize a maximal invariant under this group in a manageable form. A permutation test based upon the 3rd quadrant layer ranks $\mathbf{L}$ is proposed in Section 2 and is shown to be unbiased. A large sample unconditional test is proposed in Section 3 as an approximation to the permutation test and it is shown to be consistent. Section 4 contains results on the asymptotic distribution under local ordered shift alternatives and the Pitman efficacy of the test. Modifications of the test for some variants of the basic problem and a parametric test under normal theory are discussed in Section 5.