Maximum Likelihood Estimation with the Two-Parameter Exponential Model and Interval-Censored Data

With exact lifetime data using maximum likelihood to estimate the scale and threshold parameters for a two-parameter exponential model is quite simple. However, in the presence of interval-censored data these calculations become much more difficult, especially when the censoring is random. In this paper we discuss the mathematics underlying the determination of the maximum likelihood estimators for both the scale and threshold parameters for the two-parameter exponential model in the presence of random interval-censored data. In addition, we prove a few theoretical results concerning the maximum likelihood estimators, results that greatly restrict the situations under which the log-likelihood function could have more than one local maximum. Finally, we present results concerning the speed and accuracy of two different methods for determining these maximum likelihood estimators numerically.