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.

[1]  Yada Zhu,et al.  Parametric Estimation for Window Censored Recurrence Data , 2014, Technometrics.

[2]  Laurens de Haan,et al.  On maximum likelihood estimation of the extreme value index , 2004, math/0407062.

[3]  M. R. Leadbetter,et al.  On Extreme Values in Stationary Random Fields , 1998 .

[4]  Armelle Guillou,et al.  Statistics of Extremes Under Random Censoring , 2006, 0803.2162.

[5]  H. Nagaraja,et al.  Fisher information in window censored renewal process data and its applications , 2011 .

[6]  Enrique E. Alvarez,et al.  Smoothed nonparametric estimation in window censored semi-Markov processes , 2005 .

[7]  Niels Keiding,et al.  Product-limit estimators of the gap time distribution of a renewal process under different sampling patterns , 2010, Lifetime data analysis.

[8]  Eric P. Smith,et al.  An Introduction to Statistical Modeling of Extreme Values , 2002, Technometrics.

[9]  Nikolaos Limnios,et al.  Nonparametric Estimation for Semi-Markov Processes Based on its Hazard Rate Functions , 1999 .

[10]  Y. Vardi,et al.  Nonparametric Estimation in the Presence of Length Bias , 1982 .

[11]  Alan F. Karr,et al.  Estimation and reconstruction for zero-one Markov processes , 1984 .

[12]  Jing Qin,et al.  Empirical likelihood-based confidence intervals for length-biased data. , 2013, Statistics in medicine.

[13]  Enrique E. Alvarez,et al.  Maximum Likelihood Estimation in Alternating Renewal Processes Under Window Censoring , 2006 .

[14]  Y. Vardi.  Empirical Distributions in Selection Bias Models , 1985 .

[15]  Jing Ning,et al.  Maximum Likelihood Estimations and EM Algorithms With Length-Biased Data , 2011, Journal of the American Statistical Association.

[16]  Paul P Jovanis,et al.  Crashes and crash-surrogate events: exploratory modeling with naturalistic driving data. , 2012, Accident; analysis and prevention.