A Semiparametric Approach to Hazard Estimation with Randomly Censored Observations

Abstract

One method for estimating the hazard function is to use a parametric estimator, provided that the underlying distribution of the data can be assumed to belong to some well known family of distributions depending on an unknown, possibly vector-valued parameter. Another approach is to use nonparametric estimators, such as Cox's proportional hazard models, kernel hazard estimators, and so on. However, nonparametric estimators are less efficient than suitably chosen parametric models. But, regardless of how suitable a parametric model may be, because of errors associated with data collection, it is impossible to determine with certainty whether the observed data are actually generated by the postulated model. Consequently, instead of using exclusively a parametric or a nonparametric estimator, we propose to fit a weighted average of both. The weight is estimated by minimizing the mean square error of the combination. The main point is that we expect the proposed model to assign more weight to the estimator that best fits the data. Indeed, we show that when the parametric model holds, the proposed hazard estimator converges to the true hazard function at the same rate as the parametric hazard estimator; otherwise, it converges at the same rate as the nonparametric estimator.