Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-27T08:50:13.606Z Has data issue: false hasContentIssue false
  • English
  • Français

Moving-average models with bivariate exponential and geometric distributions

Published online by Cambridge University Press:  14 July 2016

Naftali A. Langberg  and
David S. Stoffer
Naftali A. Langberg*
Affiliation:
University of Haifa
David S. Stoffer*
Affiliation:
University of Pittsburgh
*
∗Postal address: Department of Statistics, University of Haifa, Mount Carmel, Haifa 31999, Israel.
∗∗Postal address: Department of Mathematics and Statistics, Faculty of Arts and Sciences, University of Pittsburgh, Pittsburgh, PA 15260, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

Two classes of finite and infinite moving-average sequences of bivariate random vectors are considered. The first class has bivariate exponential marginals while the second class has bivariate geometric marginals. The theory of positive dependence is used to show that in various cases the two classes consist of associated random variables. Association is then applied to establish moment inequalities and to obtain approximations to some joint probabilities of the bivariate processes.

Keywords

ASSOCIATIONBIVARIATE MOVING AVERAGESBIVARIATE POINT PROCESSES
Type
Research Papers
Information
Journal of Applied Probability , Volume 24 , Issue 1 , March 1987 , pp. 48 - 61
Copyright
Copyright © Applied Probability Trust 1987 

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 partially by the Air Force Office of Scientific Research under Contract AFOSR-84-0113 at the University of Pittsburgh.

Supported partially by the Air Force Office of Scientific Research under Contracts F49620–K-0001 and AFOSR-84-0113.

References

Arnold, B. C. (1975) A characterization of the exponential distribution by multivariate geometric compounding. Sankhya A 37, 164–173.Google Scholar
Barlow, R. E. and Proschan, F. (1975) Statistical Theory of Reliability and Life-Testing: Probability Models. Holt, Rinehart and Winston, New York.Google Scholar
Block, H. W. (1977) A family of bivariate life distributions. In The Theory and Applications of Reliability, Vol. I., Eds. Tsokos, C. P. and Shimi, I. Academic Press, New York.Google Scholar
Block, H. W. and Paulson, A. S. (1984) A note on infinite divisibility of some bivariate exponential geometric distributions arising from a compounding process. Sankhya A 46, 102–109.Google Scholar
Esary, J. D. and Marshall, A. W. (1974) Multivariate distributions with exponential minimums. Ann. Statist. 2, 84–96.10.1214/aos/1176342615Google Scholar
Esary, J. D., Proschan, F. and Walkup, D. W. (1967) Association of random variables with replications. Ann. Math. Statist. 38, 1466–1474.10.1214/aoms/1177698701Google Scholar
Downton, F. (1970) Bivariate exponential distributions in reliability theory. J. R. Statist. Soc. B 32, 408–417.Google Scholar
Freund, J. (1961) A bivariate extension of the exponential distribution. J. Amer. Statist. Assoc. 56, 971–977.10.1080/01621459.1961.10482138Google Scholar
Gaver, D. P. and Lewis, P. A. W. (1980) First-order autoregressive gamma sequences and point processes. Adv. Appl. Prob. 12, 727–745.10.2307/1426429Google Scholar
Gumbel, E. J. (1960) Bivariate exponential distributions J. Amer. Statist. Assoc. 55, 698–707.10.1080/01621459.1960.10483368Google Scholar
Hannan, E. J. (1970) Multiple Time Series. Wiley, New York.10.1002/9780470316429Google Scholar
Hawkes, A. G. (1972) A bivariate exponential distribution with applications to reliability. J. R. Statist. Soc. B 34, 129–131.Google Scholar
Jacobs, P. A. (1978) A closed cyclic queuing network with dependent exponential service times. J. Appl. Prob. 15, 573–589.10.2307/3213120Google Scholar
Jacobs, P. A. and Lewis, P. A. W. (1977) A mixed autoregressive-moving average exponential sequence and point process (EARMA(1, 1)). Adv. Appl. Prob. 9, 87–104.10.2307/1425818Google Scholar
Jacobs, P. A. and Lewis, P. A. W. (1978a) Discrete time series generated by mixtures I: Correlational and runs properties. J. R. Statist. Soc. B 40, 94–105.Google Scholar
Jacobs, P. A. and Lewis, P. A. W. (1978b) Discrete time series generated by mixtures II: Asymptotic properties. J. R. Statist. Soc. B 40, 222–228.Google Scholar
Jacobs, P. A. and Lewis, P. A. W. (1983) Stationary discrete autoregressive moving average time series generated by mixtures. J. Time Series Anal. 4, 18–36.10.1111/j.1467-9892.1983.tb00354.xGoogle Scholar
Langberg, N. A. and Stoffer, D. S. (1985) Moving average models with geometric marginals. Technical report. To appear.Google Scholar
Lawrance, A. J. and Lewis, P. A. W. (1977) A moving average exponential point process (EMA1). J. Appl. Prob. 14, 98–113.10.2307/3213263Google Scholar
Lawrance, A. J. and Lewis, P. A. W. (1980) The exponential autoregressive-moving average process EARMA(p, q). J. R. Statist. Soc. B 42, 150–161.Google Scholar
Lehmann, E. L. (1966) Some concepts of dependence. Ann. Math. Statist. 37, 1137–1153.10.1214/aoms/1177699260Google Scholar
Lewis, P. A. W. (1980) Simple models for positive-valued and discrete-valued time series with ARMA correlation structure. Multivariate Analysis V, ed. Krishnaiah, P. R., North-Holland, Amsterdam, 151–166.Google Scholar
Marshall, A. W. and Olkin, I. (1967) A multivariate exponential distribution. J. Amer. Statist. Assoc. 62, 30–44.10.1080/01621459.1967.10482885Google Scholar
Paulson, A. S. (1973) A characterization of the exponential distribution and a bivariate exponential distribution. Sankhya A 35, 69–78.Google Scholar
Tong, Y. L. (1980) Probability Inequalities in Multivariate Distributions. Academic Press, New York.Google Scholar