Estimation and efficiency with recurrent event data under informative monitoring

This article deals with studies that monitor occurrences of a recurrent event for n subjects or experimental units. It is assumed that the i  th unit is monitored over a random period [0,τi][0,τi]. The successive inter-event times Ti1,Ti2,…,Ti1,Ti2,…, are assumed independent of τiτi. The random number of event occurrences over the monitoring period is Ki=max{k∈{0,1,2,…}:Ti1+Ti2+⋯+Tik≤τi}Ki=max{k∈{0,1,2,…}:Ti1+Ti2+⋯+Tik≤τi}. The TijTij's are assumed to be i.i.d. from an unknown distribution function F   which belongs to a parametric family of distributions C={F(·;θ):θ∈Θ⊂Rp}C={F(·;θ):θ∈Θ⊂Rp}. The τiτi's are assumed to be i.i.d. from an unknown distribution function G  . The problem of estimating θθ, and consequently the distribution F  , is considered under the assumption that the τiτi's are informative about the inter-event distribution. Specifically, 1-G=(1-F)β1-G=(1-F)β for some unknown β>0β>0, a generalized Koziol–Green [cf., Koziol, J., Green, S., 1976. A Cramer–von Mises statistic for randomly censored data. Biometrika 63, 139–156; Chen, Y., Hollander, M., Langberg, N., 1982. Small-sample results for the Kaplan–Meier estimator. J. Amer. Statist. Assoc. 77, 141–144] model. Asymptotic properties of estimators of θθ, ββ, and F   are presented. Efficiencies of estimators of θθ and F are ascertained relative to estimators which ignore the informative monitoring aspect. These comparisons reveal the gain in efficiency when the informative structure of the model is exploited. Concrete demonstrations were performed for F exponential and a two-parameter Weibull.