An adapted discrete Lindley model emanating from negative binomial mixtures for autoregressive counts

Analysing autoregressive counts over time remains a relevant and evolving matter of interest, where oftentimes the assumption of normality is made for the error terms. In the case when data are discrete, the Poisson model may be assumed for the structure of the error terms. In order to address the equidispersion restriction of the Poisson distribution, various alternative considerations have been investigated in such an integer environment. This paper, inspired by the integer autoregressive process of order 1, incorporates negative binomial shape mixtures via a compound Poisson Lindley model for the error terms. The systematic construction of this model is offered and motivated, and is analysed comparatively against common alternate candidates with a number of simulation and data analyses. This work provides insight into noncentral-type behaviour in both the continuous Lindley model and in the discrete case for meaningful application and consideration in integer autoregressive environments.The National Research Foundation (NRF) of South Africa; the RDP296/2022 grant from the University of Pretoria, South Africa; the Department of Library Services based at the University of Pretoria; the University Capacity Development; and the Centre of Excellence in Mathematical and Statistical Sciences based at the University of the Witwatersrand, Johannesburg, South Africa.https://www.mdpi.com/journal/mathematicsam2023Statistic