Efficient estimation of the marginal mean of recurrent events

Recurrent events are often encountered in clinical and epidemiological studies where a terminal event is also observed. With recurrent events data it is of great interest to estimate the marginal mean of the cumulative number of recurrent events experienced prior to the terminal event. The standard nonparametric estimator was suggested in Cook and Lawless and further developed in Ghosh and Lin. We here investigate the efficiency of this estimator that, surprisingly, has not been studied before. We rewrite the standard estimator as an inverse probability of censoring weighted estimator. From this representation we derive an efficient augmented estimator using efficient estimation theory for right-censored data. We show that the standard estimator is efficient in settings with no heterogeneity. In other settings with different sources of heterogeneity, we show theoretically and by simulations that the efficiency can be greatly improved when an efficient augmented estimator based on dynamic predictions is employed, at no extra cost to robustness. The estimators are applied and compared to study the mean number of catheter-related bloodstream infections in heterogeneous patients with chronic intestinal failure who can possibly die, and the efficiency gain is highlighted in the resulting point-wise confidence intervals.