Hostname: page-component-76fb5796d-9pm4c Total loading time: 0 Render date: 2024-04-29T13:16:15.090Z Has data issue: false hasContentIssue false
  • English
  • Français

First-order autoregressive models for gamma and exponential processes

Published online by Cambridge University Press:  14 July 2016

C. H. Sim
C. H. Sim*
Affiliation:
University of Malaya
*
∗Postal address: Department of Mathematics, Faculty of Science, University of Malaya, 59100 Kuala Lumpur, Malaysia.
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we propose an autoregressive representation for a particular type of stationary Gamma(θ–1, v) process whose n-dimensional joint distributions have Laplace transform |In + θSnVn|–v, where Sn = diag(s1, · ··, sn), Vn is an n × n positive definite matrix with elements υ ij = p|i–j|i2, i, j = 1, ···, n and p is the lag-1 autocorrelation of the gamma process. We also generalize the two-parameter NEAR(1) model of Lawrance and Lewis (1981) to an exponential first-order autoregressive model with three parameters. The correlation structure and higher-order properties of the two proposed models are also given.

Keywords

CORRELATION STRUCTURESSTOCHASTIC DIFFERENCE EQUATION
Type
Research Papers
Information
Journal of Applied Probability , Volume 27 , Issue 2 , June 1990 , pp. 325 - 332
Copyright
Copyright © Applied Probability Trust 1990 

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.)

References

Al-Osh, M. A. and Alzaid, A. A. (1987) First-order integer-valued autoregressive (INAR(1)) process. J. Time Series Anal. 8, 261–275.Google Scholar
Box, G. E. P. and Cox, D. R. (1964) An analysis of transformations. J. R. Statist. Soc. B 26, 211–252.Google Scholar
Dewald, L. S. and Lewis, P. A. W. (1985) A new Laplace second-order autoregressive time series model – NLAR(2). IEEE Trans. Inf. Theory 31, 645–651.Google Scholar
Gaver, D. P. and Lewis, P. A. W. (1980) First-order autoregressive gamma sequences and point processes. Adv. Appl. Prob. 12, 727–745.Google Scholar
Granger, C. W. J. and Anderson, A. P. (1978) An Introduction to Bilinear Time Series Models. Vandenhoeck und Ruprecht.Google Scholar
Griffiths, R. C. (1970) Infinitely divisible multivariate gamma distribution. Sankhya A 32, 393–404.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, 19–36.Google Scholar
Klimko, L. A. and Nelson, P. I. (1978) On conditional least squares estimation for stochastic process. Ann. Statist. 6, 629–642.Google Scholar
Kotz, S. and Adams, J. W. (1964) Distribution of a sum of identically distributed exponentially correlated gamma-variables. Ann. Math. Stastist. 35, 277–283.Google Scholar
Lampard, D. G. (1968) A stochastic process whose intervals between events form a first order Markov chain-I. J. Appl. Prob. 5, 648–668.Google Scholar
Lawrance, A. J. and Lewis, P. A. W. (1980) The exponential autoregressive-moving average EARMA(p, q) process. 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(1)). Adv. Appl. Prob. 13, 826–845.CrossRefGoogle Scholar
Lawrance, A. J. and Lewis, P. A. W. (1987) Higher-order residual analysis for non-linear time series with autoregressive correlation structures. Int. Statist. Rev. 55, 21–35.Google Scholar
Lloyd, E. H. and Saleem, S. D. (1979) A note on seasonal Markov chains with gamma or gamma-like distributions. J. Appl. Prob. 16, 117–128.Google Scholar
Mckenzie, E. (1986) Autoregressive-moving average processes with negative binomial and geometric marginal distributions. Adv. Appl. Prob. 18, 679–705.Google Scholar
Nelson, H. (1976) The Use of Box-Cox Transformations in Economic Time Series: An Empirical Study. Ph.D. Thesis, Economics Department, University of California, San Diego.Google Scholar
Phatarfod, R. M. (1971) Some approximate results in renewal and dam theories. J. Austral. Math. Soc. 12, 425–432.Google Scholar
Sim, C. H. (1986) Simulation of Weibull and gamma autoregressive stationary process. Commun. Statist. B 15, 1141–1146.Google Scholar
Sim, C. H. (1987) A mixed gamma ARMA(1, 1) model for river flow time series. Water Resources Res. 23, 32–36.Google Scholar
Weiss, G. (1975) Time reversibility of linear stochastic processes. J. Appl. Prob. 12, 143–171.Google Scholar