On the Identifiability of Multivariate Life Distribution Functions

Abstract

Let $(T_1, T_2)$ and $(L_1, L_2)$ be two independent bivariate random vectors with distributions $F$ and $H$. Let $\tau_1 = \min(T_1, L_1), \tau_2 = \min(T_2, L_2)$ and let $G_{0,0}(s, t) = P\{\tau_1 \leq s, \tau_2 \leq t, T_1 \leq L_1, T_2 \leq L_2\}$, $G_{0,1}(s, t) = P\{\tau_1 \leq s, \tau_2 \leq t, T_1 \leq L_1, L_2 < T_2\}, \leq L_2\}, G_{0, 1}(s, t) = P\{\tau_1 \leq s, \tau_2 \leq t, L_1 < T_1, T_2 \leq L_2\}$ and $G_{1,1}(s, t) = P\{\tau_1 \leq s, \tau_2 \leq t, L_1 < T_1, L_2 < T_2\}$. Under mild conditions the distributions $F$ and $H$ are expressed explicitly as functionals of $G_{0,0}, G_{0,0}, G_{1,0}$ and $G_{1,1}$. Necessary and sufficient conditions for the formulas to hold even when $(T_1, T_2)$ and $(L_1, L_2)$ are not independent are derived. Numerous applications are indicated. Extension of the results to $p$-dimensional distributions $(p > 2)$ is given.