Understanding the distribution of an event duration time is essential in many studies. The exact time to the event is often unavailable, and thus so is the full event duration. By linking relevant longitudinal measures to the event duration, we propose to estimate the duration distribution via the first-hitting-time model (e.g. Lee and Whitmore in Stat Sci 21(4):501–513, 2006). The longitudinal measures are assumed to follow a Wiener process with random drift. We apply a variant of the MCEM algorithm to compute likelihood-based estimators of the parameters in the longitudinal process model. This allows us to adapt the well-known empirical distribution function to estimate the duration distribution in the presence of missing time origin. Estimators with smooth realizations can then be obtained by conventional smoothing techniques. We establish the consistency and weak convergence of the proposed distribution estimator and present its variance estimation. We use a collection of wildland fire records from Alberta, Canada to motivate and illustrate the proposed approach. The finite-sample performance of the proposed estimator is examined by simulation. Viewing the available data as interval-censored times, we show that the proposed estimator can be more efficient than the well-established Turnbull estimator, an alternative that is often applied in such situations.

在许多研究中，了解事件持续时间的分布是必不可少的。事件的确切时间通常不可用，因此完整的事件持续时间也是如此。通过将相关的纵向测量与事件持续时间联系起来，我们建议通过首次命中时间模型来估计持续时间分布（例如 Lee 和 Whitmore 在 Stat Sci 21(4):501–513, 2006 中）。假定纵向测量遵循具有随机漂移的维纳过程。我们应用 MCEM 算法的变体来计算纵向过程模型中基于似然的参数估计量。这使我们能够适应众所周知的经验分布函数来估计存在缺失时间原点的持续时间分布。然后可以通过传统的平滑技术获得具有平滑实现的估计量。我们建立了所提出的分布估计器的一致性和弱收敛性，并给出了它的方差估计。我们使用来自加拿大艾伯塔省的一系列野火记录来激励和说明所提议的方法。通过模拟检验了所提出的估计器的有限样本性能。将可用数据视为区间删失时间，我们表明建议的估计量比完善的 Turnbull 估计量更有效，这是一种经常应用于这种情况的替代方案。