Conclusions - A compound (mixed) Poisson distribution is sometimes used as an alternative to the Poisson distribution for count data. Such a compound distribution, which has a negative binomial form, occurs when the population consists of Poisson distributed individuals, but with intensities which have a gamma distribution. A similar situation can occur with a repairable system when failure intensities of each system are different. A more general situation is considered where the system failures are distributed according to nonhomogeneous Poisson processes having Power Law intensity functions with gamma distributed intensity parameter. If the failures of each system in a population of repairable systems are distributed according to a Power Law process, but with different intensities, then a compound Power Law process provides a suitable model. A test, based on the ratio of the sample variance to the sample mean of count data from s-independent systems, provides a convenient way to determine if a compound model is appropriate. When a compound Power Law model is indicated, the maximum likelihood estimates of the shape parameters of the individual systems can be computed and homogeneity can be tested. If equality of the shape parameters is indicated, then it is possible to test whether the systems are homogeneous Poisson processes versus a nonhomogeneous alternative. If deterioration within systems is suspected, then the alternative in which the shape parameter exceeds unity would be appropriate, while if systems are undergoing reliability growth the alternative would be that the shape parameter is less than unity. The other parameters can also be estimated by maximum likelihood. If the test for a compound Power Law model does not reject, then the joint maximum likelihood estimates of the compound model may be unstable, or may even fail to exist, except possibly in a specified limiting sense. When this happens, an ordinary Power Law process provides a more reasonable model. Of course, this would also include the possibility of the simpler homogeneous Poisson process. Copyright © 1987 by The Institute of Electrical and Electronics Engineers, Inc.


Geosciences and Geological and Petroleum Engineering

Second Department

Mathematics and Statistics

Keywords and Phrases

Compound model; Maximum likelihood estimation; Mixed model; Power-law process

International Standard Serial Number (ISSN)

1558-1721; 0018-9529

Document Type

Article - Journal

Document Version


File Type





© 2023 Institute of Electrical and Electronics Engineers, All rights reserved.

Publication Date

01 Jan 1987