Asymptotic Theory of Likelihood Ratio and Rank Order Tests in Some Multivariate Linear Models

Abstract

The purpose of this paper is two-fold: (i) to develop the asymptotic distribution theory of the normal theory likelihood ratio test statistic for the (multivariate) general linear hypothesis problem when the parent distribution is not necessarily normal and (ii) to develop the theory of the multivariate analysis of covariance based on general rank scores. The problem (i) extends the distribution theory of the likelihood ratio statistic developed by the authors in [9] for the multivariate general linear hypothesis problem (for a class of simple alternatives) to the more general case where one has also to deal with a set of concomittant variables, and the problem (ii) extends the results of the authors' earlier paper [8] on the rank order theory of the univariate analysis of covariance to the corresponding multivariate case. Let $\mathbf{Z}_{k\alpha} = (\mathbf{Y}_{k\alpha}, \mathbf{X}_{k\alpha})$; \lbrack where $\mathbf{Y}_{k\alpha} = (Y^{(1)}_{k\alpha}, \cdots, Y^{(p)}_{k\alpha}) \text{and} \mathbf{X}_{k\alpha} = (X^{(1)}_{k\alpha}, \cdots, X^{(q)}_{k\alpha}), p, q \geqq 1 \rbrack, \alpha = 1, \cdots, n_k$ be $n_k$ independent and identically distributed random vectors (i.i.d.r.v.) having a $(p + q)$-variate continuous cumulative distribution function (cdf) $G_k(\mathbf{z}), \mathbf{z} \in R^{p+q}$, for $k = 1, \cdots, c$. It is assumed that $\mathbf{Z}_{11}, \cdots, \mathbf{Z}_{cn_c}$ are mutually independent. Let us denote by $F_k^{(1)}(\mathbf{x})$ the (marginal) joint cdf of $\mathbf{X}_{k\alpha}$, and let $F_k^{(2)}(\mathbf{y}\mid\mathbf{x})$ be the conditional cdf of $\mathbf{Y}_{k\alpha}$, given $\mathbf{X}_{k\alpha} = \mathbf{x}, k = 1, \cdots, c$. As in the univariate theory (cf. [8, 10]), we assume that \begin{equation*} \tag{1.1} F_1^{(1)}(\mathbf{x}) = \cdots = F_c^{(1)}(\mathbf{x}),\quad\mathbf{x} \in R^q\end{equation*} and frame the null hypothesis as \begin{equation*} \tag{1.2} H_0:F_1^{(2)}(\mathbf{y}|\mathbf{x}) = \cdots = F_c^{(2)}(\mathbf{y}|\mathbf{x}).\end{equation*} We may note that under the usual additive model, viz., \begin{equation*} \tag{1.3} F_k^{(2)}(\mathbf{y}|\mathbf{x}) = F^{(2)}(\mathbf{y} - \mathbf{\tau}\_k|\mathbf{x}), \tau_k = (\tau_k^{(1)}, \cdots, \tau_k^{(p)}),\end{equation*} $k = 1, \cdots, c$, the null hypothesis $H_0$ in (1.2) implies that $\tau_1 = \cdots = \tau_c$. We are interested in the set of alternatives that (1.2) does not hold, which under the model (1.3) implies that not all $\tau_k, k = 1, \cdots, c$ are identical. The problem of multivariate analysis of covariance (MANOCA) can be viewed as a special case of the general linear hypothesis problem, considered in Anderson (1958, chapter 8). Two problems arise in this context: (i) how the likelihood ratio (1.r.) test behaves when the parent distribution is not necessarily normal, and (ii) how the multivariate generalizations of the tests considered in [8, 10] compare with the normal theory l.r. test. The purpose of the present investigation is to study these problems thoroughly.