Product-limit estimation for length-biased censored data

Length-biased and censored data may appear when analyzing times of duration. In this work, a new empirical curve F for approximating a distribution function F tinder right-censoring and length-bias is introduced. The proposed estimate is (not equal to but) closely related to the product-limit Kaplan-Meier estimator. Strong consistency and distributional convergence is established for a general empirical parameter (gamma) over tilde = g(integral phi(1)d (F) over tilde, ..., integral phi(r)d (F) over tilde). As applications, one can obtain the corresponding large sample results for estimates of the distribution function, the cumulative hazard function, and the mean residual time function. The new method is illustrated with real data concerning unemployment duration.