New One-Parameter Over-Dispersed Discrete Distribution and Its Application to the Nonnegative Integer-Valued Autoregressive Model of Order One

Count data arise in inference, modeling, prediction, anomaly detection, monitoring, resourceallocation, evaluation, and performance measurement. This paper focuses on a one-parameterdiscrete distribution obtained by compounding the Poisson and new X-Lindley distributions. Theprobability-generating function, moments, skewness, kurtosis, and other properties are derived inthe closed form. The maximum likelihood method, method of moments, least squares method, andweighted least squares method are used for parameter estimation. A simulation study is carried out.The proposed distribution is applied as the innovation in an INAR(1) process. The importance of theproposed model is confirmed through the analysis of two real datasets.