BIVARIATE DISTRIBUTION AND HAZARD FUNCTIONS WHEN A COMPONENT IS RANDOMLY TRUNCATED

In random truncation models one observes the i.i.d. pairs (T-i less th an or equal to Y-i), i=l,...,n. If Y is the variable of interest, then T is another independent variable which prevents the complete observa tion of Y and random left truncation occurs. Such a type of incomplete data is encountered in medical studies as well as in economy, astrono my, and insurance applications. Let (Y, Y) be a bivariate vector of ra ndom variables with joint distribution function F(y, x) and suppose th e variable Y is randomly truncated From the left. In this study, nonpa rametric estimators for the bivariate distribution and hazard function s are considered. A nonparametric estimator for F(y, x) is proposed an d an a.s. representation is obtained. This representation is used to e stablish the consistency and the weak convergence of the empirical pro cess. 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 b ehaviors of the random variables in opposite ''time'' directions. Esti mators for this vector are presented and the large sample properties a re discussed. Possible applications and a moderate size simulation stu dy are also presented. (C) 1997 Academic Press.