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

Abstract

Let (X, Y) be a bivariate random vector with joint distribution function F-X,(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 (X-i, Y-i), i = 1, 2,..., n is a random sample from (X, Y) but we are able to observe only the data consisting of those pairs (X-i, Y-i) for which X-i <= Y-i. We denote such pairs as (X-i*, Y-i*), i = 1, 2,...,v, where. is a random variable. The main problem of interest is to express the distribution function F-X,(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*, Y-i*). The copula of bivariate random vector (X*, Y*) is also derived.