Magnitudinal Effects in the Normal Multivariate Model

Abstract

Suppose the $(k \times 1)$ vectors $\mathbf{x}$ and $\mathbf{y}$ are independent with $\mathbf{x} \sim N(\mu, \Sigma)$ and $\mathbf{y} \sim N(\eta, \Sigma), \Sigma$ positive definite. If for a positive scalar $c, \mathbf{\eta} = c\mu$, we find the posterior of $c$, using noninformative priors, given the data $\{\mathbf{x}_i\}^{N_1}_1, \{\mathbf{y}_j\}^{N_2}_1$. The $\mathbf{x}_i$ are $N_1$ independent observations on $\mathbf{x}$, and independent of the $\mathbf{y}_j$, which are $N_2$ independent observations on $\mathbf{y}$. The constant $c$ is called the magnitudinal effect, and the posterior of $c$ turns out to involve a truncated Student-$t$ kernel. We also discuss the situation in which we wish to examine the truth of the statement $\mathbf{\eta} = c\mathbf{\mu}$, and proceed as follows. We first note that the matrix $\Lambda = (\lambda_{ij})$, where $\lambda_{11} = N_1\mathbf{\mu'}\Sigma^{-1}\mathbf{\mu}, \lambda_{12} = \lambda_{21} = \sqrt{N_1N_2}\mathbf{\mu'}\Sigma^{-1} \mathbf{\eta}$, and $\lambda_{22} = N_2\mathbf{\eta'}\Sigma^{-1}\mathbf{\eta}$, has a zero eigenroot if and only if $\mathbf{\eta} = c\mathbf{\mu}$, for some $c$. Hence, we are motivated to find the (joint) posterior distribution of $\omega_1, \omega_2$, the roots of $\Lambda$, where $\omega_1 > \omega_2 \geq 0$. Then, by integration with respect to $\omega_1$ over the region $\omega_1 > \omega_2$, we may find the marginal of $\omega_2$, and use it to examine the statement $\mathbf{\eta} = c\mathbf{\mu}$. The posterior of $(\omega_1, \omega_2)$ involves the multivariate hypergeometric function $_1F_^{(2)}1$, which in practice creates computational difficulties. Accordingly, some numerical considerations are discussed for computing of the posterior of $\omega_2$, and an example using real data is given.