Nonparametric analysis of doubly truncated data

Abstract

Double truncation, Nonparametric MLE, Kendall’s tau,

DTDA: AnRPackage to Analyze Randomly Truncated Data

“…The quantity F i will represent the amount of mass contributed by the lifetime DF on the truncation interval [U i , V i ]. As noted by Shen (2008), the full likelihood, L(f, k), can be decomposed as a product of the conditional likelihood of the X i 's given the (U i , V i )'s, say L 1 (f ), and the marginal likelihood of the (U i , V i )'s, say L 2 (f, k):…”

“…This will be the case, for example, when analyzing the truncation pattern, which may be informative about different features of the process under investigation. The problem of estimating the DF of the truncation times was first discussed by Shen (2008), who provided an algorithm to jointly compute the DF of both the lifetime and the truncation random variables.…”

“…In order to introduce Shen (2008) algorithm, interchange the roles of the X i 's and the (U i , V i )'s in the decomposition of equation (2). Hence, the full likelihood can be also written as the product…”

“…Maximization of L 1 (k) leads to ak = argmax k L 1 (k) such that: Shen (2008) proved that the solutions to equations (4) and (8) are not only the conditional but also the unconditional NPMLE's of F and K respectively, and that both estimators can be obtained in a simultaneous way by solving the following two equations, for j = 1, . .…”

“…This package contains tools for performing three different but related algorithms to compute the nonparametric maximum likelihood estimator of the survival function in the presence of random truncation. More precisely, the package implements the algorithms proposed by Efron and Petrosian (1999) and Shen (2008), for analyzing randomly one-sided and two-sided (i.e., doubly) truncated data. These algorithms and some recent extensions are briefly reviewed.…”