Abstract

Non negative Integer-valued auto regressive (INAR) models have been widely used for modelling the count time series data. These models have shown promising applicability in various fields such as health, insurance, and marketing etc. Number of daily/weekly cases of a disease, weekly number of insurance claims, number of items sold per day of a particular product are some of the examples of the count time series which can be modeled using INAR models. Many times such series can exhibit the seasonality. Various models have been proposed for the non seasonal count time series data, but very few have been proposed for the seasonal count time series data. We propose a INAR(1) process with seasonality for dealing with count time series with deflation or inflation of zeros. The proposed model is also capable of capturing under dispersion and over dispersion which sometimes are caused by deflation or inflation of zeros. We forecasting seasonal zero modified geometric integer valued autoregressive process of order 1 or ZMGINAR(1)s with Geometric marginal distribution. In the context of an over dispersed or under dispersed count time series data, we consider the seasonal ZMGINAR(1) and study the k-step ahead forecasting distribution corresponding to this process in detail using probability generating function. When an integer valued time series is over dispersed, Poisson time series model may not be a good choice. McKenzie (1986) proposed the INAR(1) process with geometric and negative binomial distribution as the marginals. When the count time series data has some large observation in the tail part, the geometric INAR(1) process and negative binomial INAR(1) process may be some suitable alternatives. Coherent forecasting, which is an integral part of count time series analysis, has got very little attention in the context of integer-valued time series analysis. Here, the coherent forecasting means forecasting values are to be integer. So far very few works on coherent forecasting have been done in the count time series context. Freeland and McCabe (2004) possibly be the first authors who used the concept of k-step ahead coherent forecasting of X n+k given the available data Xn , Xn−1, . . . , X1 of the time series process by using the median and mode of the k-step ahead forecasting distribution. Although the mean of a discrete distribution may not be an integer, median and mode are always so. Moreover, median has optimizing properties like it minimizes the expected absolute error E. On the other hand, mode has properties like k-step ahead forecasting distribution attains its maximum value at it. We consider the seasonal zero modified geometric integer valued autoregressive process of order 1 or ZMGINAR(1)s and study its coherent forecasting with some extensive simulation study.