This project studies the reserving problem for incurred but not reported (IBNR) claims in non-life insurance. Based on an idea presented in Kremer (1995), we propose a new Poisson INAR (integer-valued autoregressive) model for the unclosed claim counts, which are the number of reported but not enough reported claims. The properties and the prediction of the proposed Poisson INAR model are discussed. We modify the estimation methods proposed in Silva et al. (2005) for the replicated INAR(1) processes to be applied to our model and introduce new algorithms for estimating the model parameters. The performance of three different estimation methods used in this project is compared, and the impact of the sample size to the accuracy of the estimates is examined in the simulation study. To illustrate, we also present the prediction results of our proposed model using a generated sample.