On Probabilities of Rectangles in Multivariate Student Distributions: Their Dependence on Correlations

Abstract

Recently several authors (cf. [5], [6], [8], [9]) have established for arbitrary positive numbers $c_1,\cdots, c_k$ the inequality \begin{equation*}\tag{1}P\{|X_1| \leqq c_1,\cdots, |X_k| \leqq c_k\} \geqq \Pi^k_{i=1} P\{|X_i| \leqq c_i\}\end{equation*} valid for a random vector $X = (X_1,\cdots, X_k)$ having a multivariate normal distribution with mean values 0 and with an arbitrary covariance matrix. A question then arises whether also an analogue to (1) for multivariate Student distributions holds true, i.e. the inequality \begin{equation*}\tag{2}P\{|X_1|/S_1 \leqq c_1,\cdots, |X_k|/S_k \leqq c_k\} \geqq \Pi^k_{i=1} P\{|X_i|/S_i \leqq c_i\}\end{equation*} where $X = (X_1,\cdots, X_k)$ is as before, while $S_i = (\sum^p_{\nu=1} Z^2_{i\nu})^{\frac{1}{2}}, i = 1,\cdots, k$, where $Z_\nu = (Z_{i\nu},\cdots, Z_{k\nu}), \nu = 1,\cdots, p$, is a random sample of $p$ vectors, which are mutually independent and independent of $X$, and each of which has, in the simplest case, the same normal distribution as $X$. More generally, the $Z_\nu$'s have some normal distributions with mean values 0 and with some covariance matrices which need not coincide with that of $X$ and even need not be identical. A certain proof of (2) was presented by A. Scott [6] but we shall give here a counterexample showing that, unfortunately, this proof is incorrect. However, if the correlations between $X_i$ and $X_j$ have the form $\lambda_i\lambda_j\rho_{ij} (i,j = 1,\cdots, k; i \neq j)$ where $|\lambda_i| \leqq 1 (i = 1,\cdots, k)$ and where $\{\rho_{ij}\}$ is any fixed correlation matrix, and if the correlations between $Z_{i\nu}$ and $Z_{j\nu}$ have the form $\tau_{i\nu}\tau_{j\nu} (i,j = 1,\cdots, k; i \neq j; \nu = 1,\cdots, p)$ where $|\tau_{i\nu}| < 1(i = 1,\cdots, k; \nu = 1,\cdots, p)$, we shall prove here that the left-hand side probability in (2) is a non-decreasing function of each $|\lambda_i|$ and each $|\tau_{i\nu}|$; therefore, in this case of a special correlation structure, (2) is indeed true. The general validity of (2) still remains an open question.