Bivariate distribution and the hazard functions when a component is randomly truncated

In random truncation models one observes the i.i.d. pairs (Ti≤Yi), i=1, ..., n. If Y is the variable of interest, then T is another independent variable which prevents the complete observation of Y and random left truncation occurs. Such a type of incomplete data is encountered in medical studies as well as in economy, astronomy, and insurance applications. Let (Y, Y) be a bivariate vector of random variables with joint distribution function F(y, x) and suppose the variable Y is randomly truncated from the left. In this study, nonparametric estimators for the bivariate distribution and hazard functions are considered. A nonparametric estimator for F(y, x) is proposed and an a.s. representation is obtained. This representation is used to establish the consistency and the weak convergence of the empirical process. An expression for the variance of the asymptotic distribution is presented and an estimator is proposed. Bivariate "diverse-hazard" vector is introduced which captures the individual and joint failure behaviors of the random variables in opposite "time" directions. Estimators for this vector are presented and the large sample properties are discussed. Possible applications and a moderate size simulation study are also presented. © 1997 Academic Press.