The Annals of Statistics

Random Truncation Models and Markov Processes

Niels Keiding and Richard D. Gill

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Random left truncation is modelled by the conditional distribution of the random variable $X$ of interest, given that it is larger than the truncating random variable $Y$; usually $X$ and $Y$ are assumed independent. The present paper is based on a simple reparametrization of the left truncation model as a three-state Markov process. The derivation of a nonparametric estimator is a distribution function under random truncation is then a special case of results on the statistical theory of counting processes by Aalen and Johansen. This framework also clarifies the status of the estimator as a nonparametric maximum likelihood estimator, and consistency, asymptotic normality and efficiency may be derived directly as special cases of Aalen and Johansen's general theorems and later work. Although we do not carry through these here, we note that the present framework also allows several generalizations: censoring may be incorporated; the independence hypothesis underlying the truncation models may be tested; ties (occurring when the distributions of $F$ and $G$ have discrete components) may be handled.

Article information

Ann. Statist., Volume 18, Number 2 (1990), 582-602.

First available in Project Euclid: 12 April 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62M05: Markov processes: estimation
Secondary: 62G05: Estimation 60G55: Point processes 60G44: Martingales with continuous parameter

Counting processes delayed entry inference in stochastic processes intensity function left truncation nonparametric maximum likelihood product integral survival analysis


Keiding, Niels; Gill, Richard D. Random Truncation Models and Markov Processes. Ann. Statist. 18 (1990), no. 2, 582--602. doi:10.1214/aos/1176347617.

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