Discrete-valued ARMA processes

This paper presents a unified framework of stationary ARMA processes for discrete-valued time series based on Pegram’s [Pegram, G.G.S., 1980. An autoregressive model for multilag markov chains. J. Appl. Probab. 17, 350–362] mixing operator. Such a stochastic operator appears to be more flexible than the currently popular thinning operator to construct Box and Jenkins’ type stationary ARMA processes with arbitrary discrete marginal distributions. This flexibility allows us to yield an ARMA model for time series of binomial or categorical observations as a special case, which was unavailable with the extended thinning operator [Joe, H., 1996. Time series models with univariate margins in the convolution-closed infinitely divisible class. J. Appl. Probab. 33, 664–677] because the binomial/categorical distribution is not infinitely divisible. We also study parameter estimation and comparison with the thinning operator based method, whenever applicable. Real data examples are used to examine and illustrate the proposed method.