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Exact distribution theory for some point process record models

Part of: Stochastic processes Distribution theory Special processes

Published online by Cambridge University Press:  01 July 2016

J. A. Bunge  and
H. N. Nagaraja
J. A. Bunge*
Affiliation:
Cornell University
H. N. Nagaraja*
Affiliation:
The Ohio State University
*
∗Postal address: Department of Economic and Social Statistics, Cornell University, Ithaca, NY 14851-0952, USA.
∗∗Postal address: Department of Statistics, The Ohio State University, Columbus, OH 43210-1247, USA.
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Abstract

Let Y0, Y1, Y2, … be an i.i.d. sequence of random variables with continuous distribution function, and let P be a simple point process on 0≦t≦∞, independent of the Yj's. We assume that P has a point at t = 0; we associate Yj with the jth point of j≧0, and we say that the Yj's occur at the arrival times of P. Y0 is considered a ‘reference value'. The first Yj (j≧1) to exceed all previous ones is called the first ‘record value', and the time of its occurrence is the first ‘record time'. Subsequent record values and times are defined analogously. We give an infinite series representation for the joint characteristic function of the first n record times, for general P; in some cases the series can be summed. We find the intensity of the record process when P is a general birth process, and when P is a linear birth process with m immigration sources we find the distribution of the number of records in (0, t]. For m = 0 (the Yule process) we give moments of record times and a compact form for the record process intensity. We show that the records occur according to a homogeneous Poisson process when m = 1, and we display a different model with the same behavior, leading to statistical non-identifiability if only the record times are observed. For m = 2, the records occur according to a semi-Markov process; again we display a different model with the same behavior. Finally we give a new derivation of the joint distribution of the interrecord times when P is an arbitrary Poisson process. We relate this result to existing work and to the classical record model. We also obtain a new characterization of the exponential distribution.

Keywords

RANDOM RECORD MODELRECORD TIMESINTERRECORD TIMESPOISSON PROCESSYULE PROCESSBIRTH PROCESSIDENTIFIABILITYCHARACTERIZATION

MSC classification

Primary: 62E15: Exact distribution theory
Secondary: 60G55: Point processes 60K15: Markov renewal processes, semi-Markov processes
Type
Research Article
Information
Advances in Applied Probability , Volume 24 , Issue 1 , March 1992 , pp. 20 - 44
Copyright
Copyright © Applied Probability Trust 1992 

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