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First-order autoregressive gamma sequences and point processes

Published online by Cambridge University Press:  01 July 2016

D. P. Gaver  and
P. A. W. Lewis
D. P. Gaver
Affiliation:
Naval Postgraduate School, Monterey
P. A. W. Lewis
Affiliation:
Naval Postgraduate School, Monterey
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Abstract

It is shown that there is an innovation process {∊n} such that the sequence of random variables {Xn} generated by the linear, additive first-order autoregressive scheme Xn = pXn-1 + ∊n are marginally distributed as gamma (λ, k) variables if 0 ≦p ≦ 1. This first-order autoregressive gamma sequence is useful for modelling a wide range of observed phenomena. Properties of sums of random variables from this process are studied, as well as Laplace-Stieltjes transforms of adjacent variables and joint moments of variables with different separations. The process is not time-reversible and has a zero-defect which makes parameter estimation straightforward. Other positive-valued variables generated by the first-order autoregressive scheme are studied, as well as extensions of the scheme for generating sequences with given marginal distributions and negative serial correlations.

Keywords

AUTOREGRESSIVE PROCESSESEXPONENTIAL DISTRIBUTIONSGAMMA DISTRIBUTIONSSERIAL CORRELATIONSMIXED EXPONENTIAL DISTRIBUTIONSZERO-DEFECTINNOVATION PROCESSEAR(1) PROCESSGAR(1) PROCESSMEAR(1) PROCESS
Type
Research Article
Information
Advances in Applied Probability , Volume 12 , Issue 3 , September 1980 , pp. 727 - 745
Copyright
Copyright © Applied Probability Trust 1980 

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