Gamma processes. (English) Zbl 0665.62090

Lewis, P. A. W.; McKenzie, E.; Hugus, D. K.

Commun. Stat., Stochastic Models 5, No. 1, 1-30 (1989).

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.

MSC:

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

Keywords:

Beta-Gamma transformation; first-order autoregressive Beta-Gamma process; Maximum likelihood estimation; moment estimators; moving average processes; time-reversible; extended autoregressive processes; nonlinear extensions; negative correlation

Citations:

Zbl 0453.60048