Least-Squares Estimator for cumulative INAR($\infty$) processes

We explore the cumulative INAR($\infty$) process, an infinite-order extension of integer-valued autoregressive models, providing deeper insights into count time series of infinite order. Introducing a novel framework, we define a distance metric within the parameter space of the INAR($\infty$) model, which improves parameter estimation capabilities. Employing a least-squares estimator, we derive its theoretical properties, demonstrating its equivalence to a norm-based metric and establishing its optimality within this framework. To validate the estimator's performance, we conduct comprehensive numerical experiments with sample sizes $T=200$ and $T=500$. These simulations reveal that the estimator accurately recovers the true parameters and exhibits asymptotic normality, as confirmed by statistical tests and visual assessments such as histograms and Q--Q plots. Our findings provide empirical support for the theoretical underpinnings of the cumulative INAR($\infty$) model and affirm the efficacy of the proposed estimation method. This work not only deepens the understanding of infinite-order count time series models but also establishes parallels with continuous-time Hawkes processes.