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

摘要

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 v 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.