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The mixed exponential solution to the first-order autoregressive model

Published online by Cambridge University Press:  14 July 2016

A. J. Lawrance
A. J. Lawrance*
Affiliation:
University of Birmingham
*
∗Postal address: Department of Mathematical Statistics, The University of Birmingham, P.O. Box 363, Birmingham B15 2TT, U.K.
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Abstract

This paper gives the distribution of the independent variable in a first-order autoregressive equation which is necessary for the modelled variable to have a mixed exponential marginal distribution. Restricting attention first to a mixture of two exponentials which is probabilistic, a solution is shown to exist over a restricted range of the first serial correlation; tabulations of the boundary values are given. The independent variable has a distribution which includes a discrete component at zero and a not necessarily probabilistic mixture of three exponentials. The marginal distribution is then considered with a negative mixture weight; this weight is shown to be lower bounded in value, but not to depend on the first serial correlation.

Keywords

FIRST-ORDER AUTOREGRESSIVE MODELSDEPENDENT MIXED EXPONENTIAL SEQUENCESSELF-DECOMPOSABILITYPOINT PROCESSESPOSITIVE TIME SERIES
Type
Short Communications
Information
Journal of Applied Probability , Volume 17 , Issue 2 , June 1980 , pp. 546 - 552
Copyright
Copyright © Applied Probability Trust 1980 

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References

Feller, W. (1971) An Introduction to Probability Theory and its Applications, Vol. 2, 2nd edn. Wiley, New York.Google Scholar
Gaver, D. G. and Lewis, P. A. W. (1980) First-order autoregressive gamma sequences and point processes. Adv. Appl. Prob. 12, 727–745.Google Scholar
Loève, M. (1963) Probability Theory, 3rd edn. Van Nostrand, New York.Google Scholar