Small sample inference for exponential survival times with heavy right-censoring

We develop a saddlepoint-based method and several generalized Bartholomew methods for generating confidence intervals about the hazard rate of exponential survival times in the presence of heavy random right-censoring. Butler’s conditional moment generating function (MGF) formula is used to derive the MGF for the hazard rate score function which provides access to a saddlepoint-based bootstrap method. MGFs also play a key role in the generalized Bartholomew methods we develop. Since heavy censoring is assumed, the possible nonexistence of the rate parameter maximum likelihood estimate (MLE) is nonignorable. The overwhelming majority of existing methods condition upon the event that the number of observed failures is non-zero (rate parameter MLE exists). With heavy censoring these methods may not be able to produce a confidence interval an appreciable percentage of times. Our proposed methods are unconditional in the sense that they can produce confidence intervals even when the hazard rate MLE does not exist. The unconditional saddlepoint method in particular defaults in a natural way to a proposed generalized Bartholomew method when the hazard rate MLE fails to exist. We find in our Monte Carlo studies that the proposed saddlepoint method outperforms the four competing Bartholomew methods in the presence of heavy censoring and small sample sizes.