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

Let f(t₁, ⋯, tₖ) be the probability density function of a vector (Y₁, ⋯, Yₖ) of nonnegative random variables. Let the multivariate failure rate (M.F.R.) r(t₁, ⋯, tₖ) be defined by the ratio f(t₁, ⋯, tₖ)/P(Yᵢ > tᵢ, i = 1, 2, ⋯, k), for tᵢ 0, i = 1, ⋯, k. It is shown that r(t₁, ⋯, tₖ) is constant if and only if the distribution of (Y₁, ⋯, Yₖ) is a mixture of exponential distributions. Analogous results hold for the nonnegative integer valued random vector with mixture being of geometric distributions.