Estimating duration distribution aided by auxiliary longitudinal measures in presence of missing time origin

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.