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Product autoregression: a time-series characterization of the gamma distribution

Published online by Cambridge University Press:  14 July 2016

Ed Mckenzie
Ed Mckenzie*
Affiliation:
University of Strathclyde
*
∗ Postal address: Department of Mathematics, University of Strathclyde, Livingstone Tower, 26 Richmond St, Glasgow G1 1XH, U.K.
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Abstract

A non-linear stationary stochastic process {Xt} is derived and shown to have the property that both the processes {Xt} and {log Xt} have the same correlation structure, viz. the Markov or first-order autoregressive correlation structure. The generation of such processes is discussed briefly and a characterization of the gamma distribution is obtained.

Keywords

NON-LINEAR AUTOREGRESSIONSELF-DECOMPOSABILITYLOGARITHMIC TRANSFORMATIONGAMMA DISTRIBUTION CHARACTERIZATION
Type
Short Communications
Information
Journal of Applied Probability , Volume 19 , Issue 2 , June 1982 , pp. 463 - 468
Copyright
Copyright © Applied Probability Trust 1982 

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Footnotes

This paper presents material on which was based a talk of the same title given at the 13th European Meeting of Statisticians, held in Brighton in September 1980.

References

Bondesson, L. (1979) A general result on infinite-divisibility. Ann. Prob. 7, 965–979.CrossRefGoogle Scholar
Chambers, J. M., Mallows, C. L. and Stuck, B. W. (1976) A method for simulating stable random variables. J. Amer. Statist. Assoc. 71, 340–344.CrossRefGoogle Scholar
Feller, W. (1970) An Introduction to Probability Theory and its Applications, Vol. 2, 2nd edn. Wiley, New York.Google Scholar
Jones, D. A. (1978) Nonlinear autoregressive processes. Proc. R. Soc. London A 360, 71–95.Google Scholar
Kanter, M. (1975) Stable densities under change of scale and total variation inequalities. Ann. Prob. 3, 697–707.CrossRefGoogle Scholar
Khatri, C. G. and Rao, C. R. (1968) Some characterizations of the gamma distribution. Sankhya A 30, 157–166.Google Scholar
Lawrance, A. J. and Kottegoda, N. T. (1977) Stochastic modelling of river flow time-series. J. R. Statist. Soc. A 140, 1–47.Google Scholar
Levy, P. (1937) Théorie de l'addition des variables aléatories. Gauthier-Villars, Paris.Google Scholar
Loeve, M. (1963) Probability Theory, 3rd edn. Van Nostrand, New York.Google Scholar
Shanbhag, D. N., Pestana, D. and Sreehari, M. (1977) Some further results in infinite divisibility. Math. Proc. Camb. Phil. Soc. 82, 289–295.CrossRefGoogle Scholar
Shanbhag, D. N. and Sreehari, M. (1977) On certain self-decomposable distributions. Z. Wahrscheinlichkeitsth. 38, 217–222.CrossRefGoogle Scholar
Thorin, O. (1977a) On the infinite divisibility of the Pareto distribution. Scand. Actuarial J. 4, 31–40.CrossRefGoogle Scholar
Thorin, O. (1977b) On the infinite divisibility of the lognormal distribution. Scand. Actuarial J. 4, 121–148.CrossRefGoogle Scholar
Tong, H. and Lim, K. S. (1980) Threshold autoregression, limit cycles and cyclical data. J. R. Statist. Soc. B 43, 245–292.Google Scholar
Widder, D. V. (1946) The Laplace Transform. Princeton University Press.Google Scholar