The strong law for the P-L estimate in the left truncated and right censored model (I)

For the model with both left truncation and right censoring, suppose all the distributions are continuous. It is proved that the sampled cumulative hazard function Lambda(n) and the product-limit estimate F-n are strong consistent. For any nonnegative measurable phi, the almost sure convergences of integral phi d Lambda(n) and integral phi dF(n) to the true values integral phi d Lambda and integral phi dF respectively are obtained. The strong consistency of the estimator for the truncation probability is proved.MathematicsSCI(E)中国科学引文数据库(CSCD)1ARTICLE3341-3481