Consider a study where the times of occurrences of a recurrent event for n units are monitored. For the ith unit, T i1, T i2, …, denote the successive event inter-occurrence times and this unit is monitored over a random period [0, τ i ] with τ i independent of the T ij s. Over this monitoring period, is the random number of event occurrences. The T ij s are independent and identically distributed (IID) from an unknown continuous distribution function F and the τ i s are IID from a distribution function G. A generalised Koziol–Green (GKG) structure wherein 1−G=(1−F)β for some β>0 is assumed. Under this model, Nelson–Aalen and product-limit type estimators of Λ=−log(1−F) and F are obtained, as well as an estimator of β. Asymptotic and small-sample properties of these estimators are obtained and the estimator of F is compared to the fully nonparametric estimator in Peña et al. [(2001), ‘Nonparametric Estimation with Recurrent Event Data’, Journal of the American Statistical Association, 96, 1299–1315] in terms of their finite-sample and asymptotic efficiency. The performance of the estimators of F are also examined when the GKG structure does not hold through a simulation study.

中文翻译：

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信息监测下的重复事件数据的半参数估计

考虑一项研究，其中监测了 n 个单位的复发事件的发生次数。对于第 i 个单元，T i1，T i2，...，表示连续事件的发生时间，并且该单元在随机周期 [0, τ i ] 内进行监控，τ i 与 T ij s 无关。在此监视期间，是事件发生的随机数。T ij 是来自未知连续分布函数 F 的独立同分布 (IID)，而 τ 是来自分布函数 G 的 IID。 广义 Koziol-Green (GKG) 结构，其中 1−G=(1−F)假设某些 β>0 的 β。在该模型下，获得了 Λ=−log(1−F) 和 F 的 Nelson-Aalen 和乘积极限型估计量，以及 β 的估计量。获得这些估计量的渐近和小样本特性，并将 F 的估计量与 Peña 等人的完全非参数估计量进行比较。[(2001), 'Nonparametric Estimation with Recurrent Event Data', Journal of the American Statistical Association, 96, 1299–1315] 在有限样本和渐近效率方面。当 GKG 结构不通过模拟研究时，还检查 F 估计量的性能。