| The presence of truncation in multivariate data sets is becoming a recognized problem in many areas of applied statistics. A nonparametric maximum likelihood estimator (NPMLE) of the underlying distribution, first published in the astronomy literature by Lynden-Bell in 1971, has recently been analyzed in the statistics literature; most notably by Woodroofe in 1985, Wang, Jewell, and Tsai in 1986, and Chao and Lo in 1988. A limitation of this useful estimator is that it relies on the independence of the variable of interest and the truncation variable.;This thesis first introduces the right-truncation model, the Lynden-Bell estimator for right-truncated data, and the weak convergence result of Woodroofe. The asymptotic optimality of the Lynden-Bell estimator in the right-truncation model is established in chapter 2 via a proof technique due to Beran. In chapter 3, it is shown how in the right-truncation model the Lynden-Bell estimator may be represented as the sample mean of independent, identically distributed (IID) random variables. This result has been previously published by Chao and Lo, but is independently derived here.;Finally, chapter 4 enlarges the applicability of Lynden-Bell estimator beyond the usual right-truncation model. It establishes a generalization of this estimator to situations where the truncation variable and the variable of interest may be dependent, but are conditionally independent in the presence of a covariate. |
