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.

本文提出了一系列方法，用于估计一类连续时间平稳遍历随机过程中0区间长度的尾部分布与完整分布。该类随机过程以交替区间的形式取0与1两个数值。本研究的设定为：对多个此类0-1过程仅在短时间窗内完成观测，因此观测得到的0区间可能为无截尾、右截尾、左截尾或双截尾形式；且在观测窗口起始时刻仍处于持续状态的0区间，其长度服从长度偏倚分布。本文构建了针对完整分布的参数条件极大似然估计量，开发了基于半参数广义帕累托模型（semiparametric generalized Pareto model）的极大似然尾部估计方法，并提出了拟合优度图方案。本文通过仿真实验研究了有限样本下的统计性质，并针对最核心的应用场景证明了估计量的渐近正态性。随后将所提方法应用于100车自然驾驶实验（100-car study）中路面外视线扫视时长的估计任务。包含估计程序的MatLab代码与仿真实验脚本的补充材料可在线获取。