Let (T₁, T₂) and (L₁, L₂) be two independent bivariate random vectors with distributions F and H. Let τ₁ = min(T₁, L₁), τ₂ = min(T₂, L₂) and let G0,0(s, t) = P\τ₁ ≤ s, τ₂ ≤ t, T₁ ≤ L₁, T₂ ≤ L₂\, G0,1(s, t) = P\τ₁ ≤ s, τ₂ ≤ t, T₁ ≤ L₁, L₂ < T₂\, ≤ L₂\, G0, 1(s, t) = P\τ₁ ≤ s, τ₂ ≤ t, L₁ < T₁, T₂ ≤ L₂\ and G1,1(s, t) = P\τ₁ ≤ s, τ₂ ≤ t, L₁ < T₁, L₂ < T₂\. Under mild conditions the distributions F and H are expressed explicitly as functionals of G0,0, G0,0, G1,0 and G1,1. Necessary and sufficient conditions for the formulas to hold even when (T₁, T₂) and (L₁, L₂) are not independent are derived. Numerous applications are indicated. Extension of the results to p-dimensional distributions $(p > 2)$ is given.
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Langberg et al. (1982) studied this question.