The INAR(1) model is a solution of many practical problems, where the last counting process is a simple case of models AR(P) and a special case when the values of this process are positive integers, in general models can be modeled according to Markov processes, and this process will be stationary and ergodic. Previous studies present this model by means of a stochastic binomial process, the Poisson process, where the previous operations constitute a solution to many practical problems. In this thesis, we deal with the Bell distribution as an alternative to the Poisson distribution for this problem, where the last one has equal expectations and variance, which loses some flexibility and information about this problem. starting with the basic concepts: Define a new Bell INAR(1) processes. We deduce each of the initial moments of the Bell distribution, invest this distribution in the processes of INAR (1), and we study the expectation and variance of this series, the stability of this series, and the study of the properties of the Markov and ergodic process. For estimations of the coefficients of this model, we use both conditional least squares and conditional maximum likelihood estimators also raise the problem of overdispersed, the effect of using the Bell distribution in this model on this problem, compare each of these models’ using models based on the Poisson distribution and Bell. For practical applications, we implement this model with two real examples used in previous studies and compare the results.
