A Bivariate $t$ Distribution

Abstract

In this note we consider the joint distribution of Student variates $(t_1, t_2)$, where $t_1$ corresponds to the $x$-observations and $t_2$ to $y$-observations from a bivariate normal distribution. No applications are suggested as Hotelling's $T^2$ is more appropriate whenever estimation of covariance matrix is necessary. Possibly on occasions, when the correlation coefficient, $\rho$, between $x$ and $y$ may be assumed known, for example from past records, the bivariate $(t_1, t_2)$ may be useful. The main interest in this distribution is theoretical. First, because this type of bivariate $(t_1, t_2)$ has never been worked out before while the joint distribution of $(\bar{x}, \bar{y}, s_1, s_2, r)$ is commonly known. Second, for degrees of freedom $n = 1$ (sample size $N = 2$) the bivariate $t$ distribution is an example of a bivariate Cauchy distribution. Lastly, the asymptotic approximation obtained in Section 3 is an application of the method of steepest descent, which has some methodological interest and can be used in other situations. There is no loss of generality, as far as the distribution of $(t_1, t_2, r)$ is concerned, in assuming the means of $x$ and $y$ to be zero and variances to be unity. The only parameter which enters into the joint distribution of $(t_1, t_2, r)$ or into that of $(t_1, t_2)$ is $\rho$. Because of the simplicity of the limiting distribution and the asymptotic approximation we will present them first, while the exact distribution are evaluated only for $n = 1$, and 3 $(N = 2$ and 4). The exact distribution for arbitrary $n$ can be worked out, in double or triple sums, following the method given for $n = 3$