Likelihood Inference for Left Truncated and Right Censored Lifetime Data

Left truncation and right censoring arise naturally in lifetime data. Left truncation arises because in many situations, failure of a unit is observed only if it fails after a certain period. Often, the units under study may not be followed until all of them fail and the experimenter may have to stop at a certain time when some of the units may still be working. This introduces right censoring into the data. Some commonly used lifetime distributions are lognormal, Weibull and gamma, all of which are special cases of the flexible generalized gamma family. Likelihood inference via the Expectation Maximization (EM) algorithm is developed to estimate the model parameters of lognormal, Weibull, gamma and generalized gamma distributions, based on left truncated and right censored data. The asymptotic variancecovariance matrices of the maximum likelihood estimates (MLEs) are derived using the missing information principle. By using the asymptotic properties of the MLEs, asymptotic confidence intervals for the parameters are constructed. For comparison purpose, Newton-Raphson (NR) method is also used for the parameter estimation, and confidence intervals based on the NR method and parametric bootstrap are also obtained. Through Monte Carlo simulations, the performance of all these methods of inference are studied. With regard to prediction analysis, the probability that a right censored unit will be working until a future year is estimated, and an asymptotic confidence interval for the probability is then derived by the delta method. All the methods of inference developed here are illustrated with some numerical examples.