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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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References

Abramowitz, M. and Stegun, I., (Ed.) (1972) Handbook of Mathematical Functions. Dover, New York.Google Scholar
Alpuim, M. (1985) Record values in populations with increasing or random dimension. Metron 43, 145–155.Google Scholar
Ballerini, R. and Resnick, S. (1987) Embedding sequences of successive maxima in extremal processes. J. Appl. Prob. 24, 827–837.Google Scholar
Bateman Manuscript Project (1953) Higher Transcendental Functions , Vol. I. McGraw-Hill, New York.Google Scholar
Blom, G., Thorburn, D. and Vessey, T. (1990) The distribution of the record position and its applications. Amer. Statistician 44, 151–153.Google Scholar
Bruss, F. T. (1988) Invariant record processes and applications to best choice modelling. Stoch. Proc. Appl. 30, 303–316.CrossRefGoogle Scholar
Bruss, F. T. and Rogers, B. (1991) Pascal processes and their characterization. Stoch. Proc. Appl. 37, 331–338.Google Scholar
Bunge, J. and Nagaraja, H. (1989) The distributions of certain record statistics from a random number of observations. Stoch. Proc. Appl. Google Scholar
Bunge, J. and Nagaraja, H. (1990) Dependence structure of Poisson paced records. Technical Report No. 450, Department of Statistics, The Ohio State University, Columbus.Google Scholar
Daley, D. and Vere-Jones, D. (1988) An Introduction to the Theory of Point Processes . Springer-Verlag, New York.Google Scholar
Davis, H. (1962) The Summation of Series. Principia Press of Trinity University, San Antonio, TX.Google Scholar
Deheuvels, P. (1981) The strong approximation of extremal processes. Z. Wahrscheinlichkeitsth. 58, 1–6.CrossRefGoogle Scholar
Deheuvels, P. (1982) Spacings, record times and extremal processes. In Exchangeability in Probability and Statistics , ed. Koch, G. and Spizzichino, F., pp. 233–243. North-Holland, Amsterdam.Google Scholar
Deheuvels, P. (1988) Strong approximations of kth records and kth record times by Wiener processes. Prob. Theory Rel. Fields 77, 195–209.CrossRefGoogle Scholar
Embrechts, P. and Omey, E. (1983) On subordinated distributions and random record processes. Math. Proc. Camb. Phil. Soc. 93, 339–353.Google Scholar
Feller, W. (1966) An Introduction to Probability Theory and Its Applications , Vol. 2. Wiley, New York.Google Scholar
Gaver, D. (1976) Random record models. J. Appl. Prob. 13, 538–547.Google Scholar
Gaver, D. and Jacobs, P. (1978) Nonhomogeneously paced random records and associated extremal processes. J. Appl. Prob. 15, 543–551.Google Scholar
Huang, W. and Chen, L. (1989) Note on a characterization of gamma distributions. Statist. Prob. Lett. 8, 485–487.Google Scholar
Kakosyan, A., Klebanov, L. and Melamed, J. (1984) Characterizations of Distributions by the Method of Intensively Monotone Operators. Lecture Notes in Mathematics 1088. Springer Verlag, New York.Google Scholar
Karlin, S. and Taylor, M. (1975) A First Course in Stochastic Processes. Academic Press, New York.Google Scholar
Karr, A. (1986) Point Processes and their Statistical Inference. Marcel Dekker, New York.Google Scholar
Kotz, S. and Steutel, F. (1988) Note on a characterization of exponential distributions. Statist. Prob. Lett. 6, 201–203.Google Scholar
Nagaraja, H. (1988) Record values and related statistics—a review. Commun. Statist.—Theory Meth. 17, 2223–2238.Google Scholar
Nevzorov, V. (1987) Records. Theory Prob. Appl. 32, 201–228.Google Scholar
Pfeifer, D. (1981) Asymptotic expansions for the mean and variance of logarithmic inter-record times. Meth. Operat. Res. 39, 113–121.Google Scholar
Pfeifer, D. (1982) Characterizations of exponential distributions by independent non-stationary record increments. J. Appl. Prob. 19, 127–135.CrossRefGoogle Scholar
Pfeifer, D. (1984) Limit laws for inter-record times for nonhomogeneous Markov chains. J. Organizational Behav. Statist. 1, 69–74.Google Scholar
Pickands Iii, J. (1971) The two dimensional Poisson process and extremal processes. J. Appl. Prob. 8, 745–756.CrossRefGoogle Scholar
Puri, P. (1985) On nonidentifiability problems among some stochastic models in reliability theory. In Proc. Berkeley Conf. in Honor of Jerzy Neyman and Jack Keifer , ed. Le Cam, L. M. and Olshen, R. A., Volume II, pp. 729–748. Wadsworth, Monterey, CA.Google Scholar
Rényi, A. (1962), (1976) On outstanding values of a sequence of observations. In Selected Papers of A. Rényi , Vol. 3, pp. 50–65. Akadémiai Kiadó, Budapest.Google Scholar
Resnick, S. (1987) Extreme Values, Regular Variation, and Point Processes. Springer-Verlag, New York.CrossRefGoogle Scholar
Shorrock, R. (1973) Record values and inter-record times. J. Appl. Prob. 10, 543–555.Google Scholar
Stromberg, K. (1981) An Introduction to Classical Real Analysis. Wadsworth, Belmont, CA.Google Scholar
Tata, ?. (1969) On outstanding values in a sequence of random variables. Z. Wahrscheinlichkeitsth. 12, 9–20.Google Scholar
Westcott, M. (1977) The random record model. Proc. R. Soc. London A 356, 529–547.Google Scholar
Westcott, M. (1979) On the tail behavior of record time distributions in a random record process. Ann. Prob. 7, 868–873.Google Scholar
Williams, D. (1973) On Rényi's ‘record’ problem and Engel's series. Bull. London Math. Soc. 5, 235–237.Google Scholar
Yang, M. (1975) On the distribution of the inter-record times in an increasing population. J. Appl. Prob. 12, 148–154.CrossRefGoogle Scholar