On Censored Bivariate Random Variables: Copula, Characterization, and Estimation

Abstract

Let (X, Y) be a bivariate random vector with joint distribution function FX, Y(x, y) = C(F(x), G(y)), where C is a copula and F and G are marginal distributions of X and Y, respectively. Suppose that (Xi, Yi), i = 1, 2, …, n is a random sample from (X, Y) but we are able to observe only the data consisting of those pairs (Xi, Yi) for which Xi ⩽ Yi. We denote such pairs as (X*i, Yi*), i = 1, 2, …, ν, where ν is a random variable. The main problem of interest is to express the distribution function FX, Y(x, y) and marginal distributions F and G with the distribution function of observed random variables X* and Y*. It is shown that if X and Y are exchangeable with marginal distribution function F, then F can be uniquely determined by the distributions of X* and Y*. It is also shown that if X and Y are independent and absolutely continuous, then F and G can be expressed through the distribution functions of X* and Y* and the stress–strength reliability P{X ⩽ Y}. This allows also to estimate P{X ⩽ Y} with the truncated observations (X*i, Yi*). The copula of bivariate random vector (X*, Y*) is also derived.