Efficient Estimation of the Lifetime and Disease Onset Distribution
We study efficient nonparametric maximum likelihood estimation of the distribution of onset and lifetime associated with an irreversible disease that is only detectable at sacrifice or death. We show that the Kaplan-Meier estimator of the lifetime distribution is asymptotically efficient, so estimation of the lifetime distribution cannot be improved by using current status information on the time until onset. The actual nonparametric maximum likelihood estimator (NPMLE) tries to use current status information on the time until onset, but it is asymptotically equivalent to the Kaplan-Meier estimator, and it is outperformed by the latter in simulations. The NPMLE of the onset distribution is shown to be an iteratively reweighted least squares estimator which can be compared with the weighted pool-adjacent-violators algorithm. Moreover, we show that it is unnecessary to iteratively estimate the weight since an initial estimate cannot be improved. This insight leads to a simple, explicit estimator which improves on the NPMLE of the onset distribution. The results are verified with simulations, and a data analysis example is provided.
Statistical Methodology | Statistical Theory
van der Laan, Mark J.; Jewell, Nicholas P.; and Peterson, Derick R., "Efficient Estimation of the Lifetime and Disease Onset Distribution" (August 1997). U.C. Berkeley Division of Biostatistics Working Paper Series. Working Paper 56.