;Truncation often occurs in lifetime data analysis, where samples are collected under certain time constraints. This thesis considers parametric inference when random samples are subject to double-truncation, i.e., both left- and right-truncations. Efron and Petrosian (1999) proposed to fit the special exponential family (SEF)
where and , for doubly-truncated data, but did not study it’s computational and theoretical properties. This thesis fills this gap. We develop computational algorithms for Newton-Raphson and fixed point iteration techniques to obtain maximum likelihood estimator (MLE) of the parameters, and then compare the performance of these two methods by simulations. To stabilize the convergence under the three-parameter SEF, we propose a randomized Newton-Raphson method. Also, we study the asymptotic properties of the MLE based on the theory of independent but not identically distributed (i.n.i.d) random variables that accommodate the heterogeneity of truncation intervals. Lifetime data from the Channing House study are used for illustration.