Tail estimation for window censored processes
Journal article, 2016

This paper develops methods to estimate the tail and full distribution of the lengths of the 0-intervals in a continuous time stationary ergodic stochastic process which takes the values 0 and 1 in alternating intervals. The setting is that each of many such 0-1 processes have been observed during a short time window. Thus the observed 0-intervals could be non-censored, right censored, left censored or doubly censored, and the lengths of 0-intervals which 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 semi-parametric 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.

100-car naturalistic driving study

Off-road glance

Traffic safety

Generalized Pareto distribution

Tail estimation

Length-biased distribution


Holger Rootzen

University of Gothenburg

Chalmers, Mathematical Sciences, Mathematical Statistics

Dmitrii Zholud

Chalmers, Mathematical Sciences, Mathematical Statistics

University of Gothenburg


0040-1706 (ISSN) 1537-2723 (eISSN)

Vol. 58 1 95-103

Areas of Advance


Subject Categories

Probability Theory and Statistics



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