Abstract This article provides a goodness-of-fit test for the distribution function or the survival function in a recurrent event setting, when the inter-event time parametric structure is estimated from the observed data. Of concern is the null hypothesis that the inter-event time distribution is absolutely continuous and belongs to a parametric family , where the q-dimensional parameter space is neither known nor specified. We proposed a Khmaladze martingale-transformed type of test (Khmaladze, 1981), adapted to recurrent events. The test statistic combines two likelihood sources of estimation to form a parametric empirical process: (1) a product-limit nonparametric maximum likelihood estimator (NPMLE; Peña et al., 2001a) that is a consistent estimator of F, say, and (2) a point process likelihood estimator (Jacod, 1974/1975). These estimators are combined to construct a Kolmogorov-Smirnov (KS) type of test (Kolmogorov 1933; Smirnov, 1933). Empirical process and martingale weak convergence frameworks are utilized for theoretical derivations and motivational justification of the proposed transformation. A simulation study is conducted for performance assessment, and the test is applied to a problem investigated by Proschan (1963) on air-conditioning failure in a fleet of Boeing 720 jets.

中文翻译：

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在存在复发事件的情况下，Khmaladze 变换的符合 ML 估计的检验

摘要 当从观测数据估计事件间时间参数结构时，本文提供了在循环事件设置中分布函数或生存函数的拟合优度检验。值得关注的是零假设，即事件间时间分布绝对连续并且属于参数族 ，其中 q 维参数空间既不知道也不指定。我们提出了一种 Khmaladze martingale 转换类型的测试（Khmaladze，1981），适用于反复发生的事件。检验统计量结合了估计的两个似然来源以形成参数经验过程：(1) 乘积限制非参数最大似然估计量 (NPMLE；Peña 等人，2001a)，它是 F 的一致估计量，例如，和 (2 ) 点过程似然估计器 (Jacod, 1974/1975)。这些估计量组合起来构建 Kolmogorov-Smirnov (KS) 类型的测试 (Kolmogorov 1933; Smirnov, 1933)。经验过程和鞅弱收敛框架被用于所提议的转换的理论推导和动机论证。进行了性能评估的模拟研究，并将该测试应用于 Proschan (1963) 研究的波音 720 喷气机机队空调故障问题。