Tail Estimation for Window-Censored Processes

This article develops methods to estimate the tail and full distribution of the lengths of the 0-intervals in a continuous time stationary ergodic stochastic process that takes the values 0 and 1 in alternating intervals. The setting is that each of many such 0–1 processes has been observed during a short time window. Thus, the observed 0-intervals could be noncensored, right-censored, left-censored, or doubly-censored, and the lengths of 0-intervals that are ongoing at the beginning of the observation window have a length-biased distribution. We exhibit parametric conditional maximum likelihood estimators for the full distribution, develop maximum likelihood tail estimation methods based on a semiparametric generalized Pareto model, and propose goodness-of-fit plots. Finite sample properties are studied by simulation, and asymptotic normality is established for the most important case. The methods are applied to estimation of the length of off-road glances in the 100-car study, a big naturalistic driving experiment. Supplementary materials that include MatLab code for the estimation routines and a simulation study are available online.

本文针对交替取0与1值的连续时间平稳遍历随机过程（continuous time stationary ergodic stochastic process），提出了其0值区间（0-intervals）长度的尾部分布与完整分布的估计方法。
本研究的设定为：在短时间窗口内对多个此类0-1随机过程进行观测。据此，观测得到的0值区间可分为无截尾、右截尾、左截尾与双截尾四类；且在观测窗口起始时刻仍处于持续状态的0值区间，其长度服从长度偏倚分布。
本文给出了完整分布的参数条件极大似然估计量，构建了基于半参数广义帕雷托模型的尾部极大似然估计方法，并提出了拟合优度图方案。本文通过模拟实验分析了有限样本下的估计性质，并针对核心应用场景证明了估计量的渐近正态性。
本文将所提方法应用于100车自然驾驶实验（100-car study）中驾驶员离路视线时长的估计。包含估计程序MatLab代码与模拟实验的补充材料可在线获取。