Hostname: page-component-8448b6f56d-mp689 Total loading time: 0 Render date: 2024-04-20T01:24:23.277Z Has data issue: false hasContentIssue false
  • English
  • Français

Random and Poisson paced record models in the Fα setup

Part of: Stochastic processes Distribution theory

Published online by Cambridge University Press:  14 July 2016

Glenn Hofmann  and
H. N. Nagaraja
Glenn Hofmann*
Affiliation:
Universidad de Concepción
H. N. Nagaraja*
Affiliation:
Ohio State University
*
∗Postal address: Departamento de Estadistica, Facultad de Ciencias Fisicas y Matematicas, Casilla 4009, Barrio Universitario, Concepción, Chile. Email address: glenn@gauss.cfm.udec.cl.
∗∗Postal address: Department of Statistics, Ohio State University, Columbus, OH 43210-1247, USA. Email address: hnn@stat.ohio-state.edu.
Get access Rights & Permissions [Opens in a new window]

Abstract

We study a random record model where the observation Xi has continuous distribution function Fαi (αi > 0) and the number of available observations is random and independent of the observations. We obtain the joint distribution of the record values and inter-record times for our model. We investigate the distribution of the number of records when the number of observations has one of the common distributions and the α's increase geometrically or linearly. A particularly interesting case arises when the observations arrive at time points paced by a Poisson point process. For this model we obtain distributional results for the inter-arrival times of records for a large class of combinations of α structures and intensity functions.

Keywords

Record timesrecord countsrecord valuesinter-record timesPoisson processrandom record modelstochastic convergence

MSC classification

Primary: 60G70: Extreme value theory; extremal processes 60G55: Point processes
Secondary: 62E15: Exact distribution theory 62E20: Asymptotic distribution theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 37 , Issue 2 , June 2000 , pp. 374 - 388
Copyright
Copyright © by the Applied Probability Trust 2000 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Supported in part by FONDECYT grant Number 1990343 of Chile.

References

Abramowitz, M., and Stegun, I. A. (eds) (1972). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York.Google Scholar
Arnold, B. C., Balakrishnan, N., and Nagaraja, H. N. (1998). Records. John Wiley, New York.Google Scholar
Ballerini, R., and Resnick, S. I. (1987). Embedding sequences of successive maxima in extremal processes. J. Appl. Prob. 24, 827–837.CrossRefGoogle Scholar
Bingham, N. H., Goldie, C. M., and Teugels, J. L. (1987). Regular Variation. Cambridge University Press.CrossRefGoogle Scholar
Bunge, J., and Nagaraja, H. N. (1991). The distribution of certain record statistics from a random number of observations. Stoch. Proc. Appl. 38, 167–183.Google Scholar
Bunge, J., and Nagaraja, H. N. (1992a). Exact distribution theory for some point process record models. Adv. Appl. Prob. 29, 587–596.Google Scholar
Bunge, J., and Nagaraja, H. N. (1992b). Dependence structure of Poisson-paced records. J. Appl. Prob. 24, 20–44.CrossRefGoogle Scholar
de Haan, L., and Verkade, E. (1987). On extreme value theory in the presence of a trend. J. Appl. Prob. 24, 62–76.Google Scholar
Deheuvels, P. (1982). Spacings, record times and extremal processes. In Exchangeability in Probability and Statistics, eds Koch, G. and Spizzichino, F. North-Holland, Amsterdam, pp. 233–243.Google Scholar
Feuerverger, A., and Hall, P. (1996). On distribution-free inference for record-value data with trends. Ann. Statist. 24, 2655–2678.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
Hofmann, G. (1997). A family of general record models. Ph.D. Dissertation, Ohio State University, Columbus, Ohio.Google Scholar
Nevzorov, V. B. (1986). The number of records in a sequence of non-identically distributed random variables. In Probability Distributions and Mathematical Statistics. Fan, Tashkent, pp. 373–388. Engl. transl.: J. Soviet Math. 1987, 2375–2382.Google Scholar
Nevzorov, V. B. (1995). Asymptotic distributions of records in nonstationary schemes. J. Statist. Plann. Inf. 45, 261–273.CrossRefGoogle Scholar
Pickands, J. (1971). The two dimensional Poisson process and extremal processes. J. Appl. Prob. 8, 745–756.CrossRefGoogle Scholar
Pollard, D. (1984). Convergence of Stochastic Processes. Springer, New York.CrossRefGoogle Scholar
Rao, C. R., and Shanbhag, D. N. (1994). Choquet–Deny Type Functional Equations with Applications to Stochastic Models. John Wiley, Chichester.Google Scholar
Smith, R. L. (1988). Forecasting records by maximum likelihood. J. Amer. Statist. Assoc. 83, 331–338.Google Scholar
Waugh, W. A. O'N. (1970). Transformations of a birth process into a Poisson process. J. R. Statist. Soc. B 32, 418–431.Google Scholar
Yang, M. C. K. (1975). On the distribution of the inter-record times in an increasing population. J. Appl. Prob. 12, 148–154.Google Scholar