Estimation of the truncation probability in the random truncation model

• 成果类型: Article
• 署名作者: He, SY; Yang, GL
• 署名单位: Peking University; University System of Maryland; University of Maryland College Park
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• 发表日期: 1998
• 页码: 1011-1027
• 关键词: product-limit estimator; random censorship; REPRESENTATIONS

摘要

Under random truncation, a pair of independent random variables X and Y is observable only if X is larger than Y. The resulting model is the conditional probability distribution H(x, y) = P[X less than or equal to x, Y less than or equal to y\X greater than or equal to Y]. For the truncation probability alpha = P[X greater than or equal to Y], a proper estimate is not the sample proportion but alpha(n) = integral G(n)(s)dF(n)(s) where F-n and G(n) are product limit estimates of the distribution functions F and G of X and Y, respectively We obtain a much simpler representation <(alpha)over cap>(n) for <(alpha)over cap>(n). With this, the strong consistency, an lid representation (and hence asymptotic normality), and a LIL for the estimate are established. The results are true for arbitrary F and G. The continuity restriction on F and G often imposed in the literature is not necessary. Furthermore, the representation <(alpha)over cap>(n) of alpha(n) facilitates the establishment of the strong law for the product limit estimates F-n and G(n).