On a Characterization of the Family of Distributions with Constant Multivariate Failure Rates

Let $f(t_1, \cdots, t_k)$ be the probability density function of a vector $(Y_1, \cdots, Y_k)$ of nonnegative random variables. Let the multivariate failure rate (M.F.R.) $r(t_1, \cdots, t_k)$ be defined by the ratio $f(t_1, \cdots, t_k)/P(Y_i > t_i, i = 1, 2, \cdots, k)$, for $t_i \geqq 0, i = 1, \cdots, k$. It is shown that $r(t_1, \cdots, t_k)$ is constant if and only if the distribution of $(Y_1, \cdots, Y_k)$ is a mixture of exponential distributions. Analogous results hold for the nonnegative integer valued random vector with mixture being of geometric distributions.