Threshold estimation in the log-gamma model

The three-parameter log-gamma distribution is a versatile lifetime model. However, it has a quite unusual property that allows it to be used as a potential threshold model: namely, that though it is not left-limited in general, it includes the shifted exponential as a special case. Thus, it is a useful bridge between skewed models that are not explicitly left-limited, on the one hand, and a true threshold model on the other. It is shown that the likelihood always has a local maximum corresponding to the exponential model, even when this is not the true model, so that the usual maximum likelihood (ML) estimator is unsatisfactory. An estimator obtained by maximising a certain spacing-modified likelihood is proposed that does not suffer from this problem. Its distributional properties are derived showing it to be as efficient as ML when this does work and moreover showing the estimators of shape and location to be hyper-efficient when the exponential model is the true model. Simulation results and three numerical examples are given contrasting the behaviour of the proposed estimator with the ML estimator