Gamma processes

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, pth-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 Gaver and Lewis. 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.