Asymptotic Efficiency of Multivariate Normal Score Test

Abstract

Let $X_\alpha = (X_{1\alpha}, X_{2\alpha}, \cdots, X_{p\alpha}), \alpha = 1, 2, \cdots, m$, and $\alpha = m + 1, \cdots, N$, be independent random samples of sizes $m$ and $n = N - m$ from continuous $p$-variate cdf $\Psi^{(1)}(\mathbf{x})$ and $\Psi^{(2)}(\mathbf{x})$ respectively. For testing $H_0:\Psi^{(1)} = \Psi^{(2)}$ against shift alternatives $\Psi^{(1)}(\mathbf{x} - \Delta) = \Psi^{(2)}(\mathbf{x})$ various tests are available in literature of which the important ones are the following: (a) the classical Hotelling's $T^2$ test; (b) the multivariate version of the Wilcoxon test proposed by Chatterjee and Sen ([4], [5]). The test statistic $W$ is a quadratic form in the vector of coordinatewise Wilcoxon statistics. (c) The multivariate version of the normal score test proposed by Bhattacharyya [1] and for the general multisample problem by Tamura [9] and Sen and Puri [8]. The test statistic $M$ is a quadratic form in coordinatewise normal score statistics. Both $M$ and $W$ are members of the class of tests (4.7) of Tamura [9]. For consideration of asymptotic relative efficiency (ARE) let $\Psi(\mathbf{x})$ denote the common cdf under $H_0$ and $\Delta_N = \delta/N^{\frac{1}{2}}, (\delta \neq 0)$ a sequence of shift alternatives tending to $H_0$ along the direction $\delta$. Using $e_{A:B}(\delta, \Psi)$ as a general notation for the Pitman efficiency of a test $A$ relative to a test $B$, which typically depends on $\Psi$ and $\delta$, we have under suitable regularity conditions (c.f. Theorem 3, Tamura [9]) \begin{equation*}\tag{1.1}e_{M:W}(\delta, \Psi) = \Delta_M\Delta^{-1}_W,\quad e_{M:T}(\delta, \Psi) = \Delta_M\Delta^{-1}_T\end{equation*} where \begin{equation*}\tag{1.2}\Delta_M = \delta\Lambda^{-1}\delta',\quad \Delta_W = \delta\Gamma^{-1}\delta',\quad\Delta_T = \delta\Sigma^{-1}\delta'\end{equation*} and $\Lambda = (\lambda_{ij}), \Gamma = (\gamma_{ij}), \Sigma = (\rho_{ij}\sigma_i\sigma_j)$ are nonsingular $p \times p$ matrices. $\Sigma$ is the covariance matrix of $\Psi$. Denoting by $\psi_i$ and $\Psi_i$ the $i$th marginal density and cdf of $\Psi$, by $\Psi_{ij}$ the joint $(ij)$th marginal cdf and by $\phi$ and $\Phi$ the density and cdf of standard normal distribution, the typical elements of $\Lambda$ and $\Gamma$ are given by: \begin{equation*}\tag{1.3}\lambda_{ij} = \rho^\ast_{ij}\theta^{-1}_i\theta^{-1}_j \text{and} \gamma_{ij} = \rho'_{ij}\gamma_j\gamma_i\end{equation*} where \begin{equation*}\tag{1.4}\theta_i = \int^\infty_{-\infty} \lbrack\psi^2_i(x) dx/\phi\{\Phi^{-1}\lbrack\Psi_i(x)\rbrack\}\rbrack,\quad \gamma_i = \lbrack(12)^{\frac{1}{2}} \int^\infty_{-\infty}\psi^2_i(x) dx\rbrack^{-1};\end{equation*} \begin{equation*}\tag{1.5}\rho^\ast_{ij} = \int^\infty_{-\infty} \int^\infty_{-\infty} \Phi^{-1} \lbrack\Psi_i(x)\rbrack\Phi^{-1}\lbrack\Psi_j(y)\rbrack d\Psi_{ij}(x, y);\end{equation*} \begin{equation*}\tag{1.6}\rho'_{ij} = 12 \int^\infty_{-\infty} \int^\infty_{-\infty} \Psi_i(x)\Psi_j(y) d\Psi_{ij}(x, y) - 3.\end{equation*} $(\rho_{ij}), (\rho'_{ij})$ and $(\rho^\ast_{ij})$ are respectively the product moment, the grade, and the normal score correlation matrices of $\Psi$. In this paper the ARE properties of the $M$ test relative to the $W$ and $T^2$ tests are studied by investigating the bounds of the efficiencies (1.1) for various important classes of multivariate distributions. Since the ARE expressions (1.1) for two sample tests are essentially of the same structure as those for their multisample analogues, no generality is lost, as far as efficiency bounds are concerned, by restricting discussions to the two sample situation. It is shown that in the class of nonsingular multivariate normal distributions the $M$ test has efficiency 1 relative to $T^2$ and its efficiency relative to $W$ exceeds 1 irrespective of the direction $\delta$. The $M$ test behaves very well when the parent distribution has marginal densities dropping down to zero discontinuously at either tail and also in gross error models when heavy tails are present in the contaminating distribution.