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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 methods are applied to the 100-car study, a big naturalistic driving experiment.

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Tail estimation for window-censored processes

Abstract

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

Main file: Contains exact expressions for residual life for gamma and Weibull distributions, additional results on asymptotic normality, and a brief description of numerical routines used throughout the analysis.

MatLab and Wolfram Mathematica scripts: Estimation algorithms, simulation study, analysis of 100-Car data, and numerical verification of asymptotic normality.

Reference

Rootzén, H. and Zholud, D. (2016). Tail estimation for window-censored processes, Technometrics, Vol. 58, No. 1, pp. 95-103.

BiBTeX

@article{RootzenZholud2016,
  Author = {Rootz\`{e}n, H. and Zholud, D.},
  Year = {2016},
  Title = {Tail Estimation for Window-Censored Processes},
  Journal = {Technometrics},
  Volume = {58},
  Number = {1},
  Pages = {95--103}
}

Update 2018

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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 methods are applied to the 100-car study, a big naturalistic driving experiment.

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