Hostname: page-component-7479d7b7d-m9pkr Total loading time: 0 Render date: 2024-07-10T01:55:43.290Z Has data issue: false hasContentIssue false
  • English
  • Français

Bivariate exponential and geometric autoregressive and autoregressive moving average models

Published online by Cambridge University Press:  01 July 2016

H. W. Block ,
N. A. Langberg  and
D. S. Stoffer
H. W. Block*
Affiliation:
University of Pittsburgh
N. A. Langberg*
Affiliation:
University of Haifa
D. S. Stoffer*
Affiliation:
University of Pittsburgh
*
∗Postal address: Department of Mathematics and Statistics, University of Pittsburgh, PA 15260, USA.
∗∗Postal address: Department of Statistics, University of Haifa, Mount Carmel, Haifa 31999, Israel.
∗Postal address: Department of Mathematics and Statistics, University of Pittsburgh, PA 15260, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

We present autoregressive (AR) and autoregressive moving average (ARMA) processes with bivariate exponential (BE) and bivariate geometric (BG) distributions. The theory of positive dependence is used to show that in various cases, the BEAR, BGAR, BEARMA, and BGARMA models consist of associated random variables. We discuss special cases of the BEAR and BGAR processes in which the bivariate processes are stationary and have well-known bivariate exponential and geometric distributions. Finally, we fit a BEAR model to a real data set.

Keywords

BIVARIATE EXPONENTIAL AND GEOMETRIC DISTRIBUTIONSBIVARIATE AUTOREGRESSIVE AND AUTOREGRESSIVE MOVING AVERAGE MODELS IN EXPONENTIAL AND GEOMETRIC RANDOM VECTORSASSOCIATIONJOINT STATIONARITY
Type
Research Article
Information
Advances in Applied Probability , Volume 20 , Issue 4 , December 1988 , pp. 798 - 821
Copyright
Copyright © Applied Probability Trust 1988 

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

Research partially supported by the Air Force Office of Scientific Research under Contract 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. (1981) Statistical Theory of Reliability and Life-Testing: Probability Models. To Begin With, Silver Spring, MD.Google Scholar
Block, H. W. (1977) A family of bivariate life distributions. In The Theory and Applications of Reliability, Vol. 1. Tsokos, C. P. and Shimi, I., Eds. 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 46, A, 102–109.Google Scholar
Downton, F. (1970) Bivariate exponential distributions in reliability theory. J. R. Statist. Soc. B 32, 408–417.Google Scholar
Esary, J. D. and Marshall, A. W. (1974) Multivariate distributions with exponential minimums. Ann. Statist. 2, 84–96.CrossRefGoogle Scholar
Esary, J. D. and Marshall, A. W. (1973) Multivariate geometric distributions generated by a cumulative damage process. Naval Postgraduate School Report NP555EY73041A.Google Scholar
Freund, J. (1961) A bivariate extension of the exponential distribution. J. Amer. Statist. Assoc. 56, 971–977.CrossRefGoogle Scholar
Gaver, D. P. and Lewis, P. A. W. (1980) Firt-order autoregressive gamma sequences and point processes. Adv. Appl. Prob. 12, 727–745.CrossRefGoogle Scholar
Gumbel, E. J. (1960) Bivariate exponential distributions. J. Amer. Statist. Assoc. 55, 698–707.CrossRefGoogle Scholar
Hannan, E. J. (1970) Multiple Time Series. Wiley, New York.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle Scholar
Langberg, N. A. and Stoffer, D. S. (1987) Moving average models with bivariate exponential and geometric distributions. J. Appl. Prob. 48–61.CrossRefGoogle Scholar
Lawrance, A. J. and Lewis, P. A. W. (1977) A moving average exponential point process (EMA1). J. Appl. Prob. 14, 98–113.CrossRefGoogle 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
Lawrance, A. J. and Lewis, P. A. W. (1981) A new autoregressive time series model in exponential variables (NEAR(l)). Adv. Appl. Prob. 13, 826–845.CrossRefGoogle Scholar
Lawrance, A. J. and Lewis, P. A. W. (1985) Modelling and residual analysis of nonlinear autoregressive time series in exponential variables (with Discussion). J. R. Statist. Soc. B 47, 165–202.Google 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.CrossRefGoogle Scholar
Paulson, A. S. (1973) A characterization of the exponential distribution and a bivariate exponential distribution. Sankhya A 35, 69–78.Google Scholar
Paulson, A. S. and Uppuluri, V. R. R. (1972) A characterization of the geometric distribution and a bivariate geometric distribution. Sankhya A 34, 88–91.Google Scholar
Raftery, A. E. (1982) Generalized non-normal time series. In Time Series Analysis: Theory and Practice 1, ed. Anderson, O. D., North-Holland, Amsterdam, 621–640.Google Scholar
Raftery, A. E. (1984) A continuous multivariate exponential distribution. Commun. Statist., Theor. Meth. A 13, 947–965.CrossRefGoogle Scholar
Sarkar, S. K. (1987) A continuous bivariate exponential distribution. J. Amer. Statist. Assoc. 82, 667–675.CrossRefGoogle Scholar
Smith, R. L. (1986) Maximum likelihood estimation for the NEAR(2) model. J. R. Statist. Soc. 48, 251–257.Google Scholar