DISCRETE VARIATE TIME SERIES Eddie McKenzie Department of Statistics & Modelling Science University of Strathclyde 15th August 2000 1 Introduction Modelling discrete variate time series is the most challenging and, as yet, least well developed of all areas of research in time series. The fact that variate values are integer renders most traditional representations of dependence either impossible or impractical. In the last two decades there have been a number of imaginative attempts to develop a suitable class of models. Our purpose here is to briefly review some of the most interesting and exciting of these. Discrete variate time series occur in many contexts, often as counts of events, objects or individuals in consecutive intervals or at consecutive points in time. Some simple examples are the numbers of accidents in a manufacturing plant each month, the numbers of patients treated by a hospital’s accident and emergency unit each hour, the numbers of fish caught in a particular area of sea each week, the numbers of busy lines in a telephone network noted every thirty minutes, and the numbers of lifts in a tall office building which are fully operational at the start of business each day. Such data may also arise from the discretization of continuous variate time series. An example of this is the reduction of daily rainfall volumes to a binary series of ones and zeros, i.e. wet and dry days. See, for example, Phatarfod and Srikanthan (1981) Simulation is an important use for models of discrete variate time series, since the need to generate sequences of dependent random variates with a particular marginal distribu- tion and correlation structure is common. Sometimes this is done for purely practical reasons, e.g. the attempt to simulate dam input more realistically, as in Phatarfod and Mardia (1973). Sometimes the reasons are more statistical, e.g. to assesss effects of serial correlation on procedures, i.e. tests, estimators, etc, developed originally for independent data. Often the motivation can be both theoretical and practical. For example, such models are a useful source of often new and interesting discrete multivariate distribu- tions. More generally, they may yield insight into the construction of such distributions. The work of Phatarfod and Mardia (1973) used a bivariate distribution introduced by Edwards and Gurland (1961) in studies of accident proneness. Both pairs of authors were less concerned with the underlying structure of the generating processes involved than with their joint distributions. However, two of the processes we shall see here, viz the Poisson and binomially distributed AR(1) processes defined by (6) and (18), respectively, are versions of theirs. 1