Gamma processes. (English) Zbl 0665.62090

The Beta-Gamma transformation is described and is used to define a very simple first-order autoregressive Beta-Gamma process, BGAR(1). Maximum likelihood estimation is discussed for this model, as well as moment estimators. The first-order structure is extended to include moving average processes and mixed first-order autoregressive, p th-order moving average processes. It is shown that these Gamma processes are time- reversible and, therefore, too narrow for general physical modelling.
A dual process to the BGAR(1) process, DBGAR(1), is introduced, as well as an iterated process which combines the Beta-Gamma process and the GAR(1) process of D. P. Gaver and P. A. W. Lewis [Adv. Appl. Probab. 12, 727-745 (1980; Zbl 0453.60048)]. Some properties of these extended autoregressive processes are derived. Several highly nonlinear extensions of these processes which produce negative correlation are given. Use of the processes to model a sequence of times between failures of a computer system is described.


62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62M09 Non-Markovian processes: estimation


Zbl 0453.60048
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