A novel bounded ℤ-valued autoregressive model with its application on crime data

Abstract

This paper tackles the challenge of autoregressive modelling for bounded Z -valued count time series, a topic that has been largely overlooked in the existing literature. To bridge this gap, we introduce the trinomial difference bounded Z -valued autoregressive (TDBZAR) model utilizing the trinomial difference thinning operator, and we explore its stochastic properties in detail. Two notable advantages of the TDBZAR model are that the incorporated trinomial difference thinning operator ensures the conditional expectation takes a linear form and allows its 1-step transition probability to be expressed as the convolution of two independent trinomial difference distributions. This significantly simplifies the computation of both the conditional least squares (CLS) and conditional maximum likelihood (CML) estimates, making them more practical and straightforward. Second, we examine the two-step CLS and CML estimates, establishing their asymptotic properties. Third, we compare the performance of these estimators through simulation studies. Finally, we apply the proposed model to a real dataset to investigate which type of crime (rape or sex offense) was more prevalent in police reports from January 1990 to December 1998.