A trinomial difference autoregressive model and its applications

Abstract

This paper considers the autoregressive modelling problem of ‐valued time series of counts, whose counting sequence consists of −1, 0 and 1. Most existing methods are based on the signed binomial thinning operator and its some extensions, in which the counting sequence consists of 0 and 1, that is, the cases of −1 is ignored. To fill this gap, we first construct the trinomial difference thinning operator and then propose the trinomial difference Z‐valued autoregressive (TDZAR) model and give some stochastic properties. An attractive merit of the TDZAR model is that the incorporated trinomial difference thinning operator makes the conditional maximum likelihood estimate more wieldy and easy. Second, we discuss the two‐step conditional least squares estimate and the conditional maximum likelihood estimate and establish their asymptotic properties of the estimators. Third, the performances of these estimators are compared via simulation study. Finally, we apply the proposed model to a real data set.

A Trinomial Difference Autoregressive Model for the Bounded Z-valued time series

“…Then αcls and βcls can be obtained by αcls = ρcls 1 + ρcls 2 /n and βcls = ρcls 2 /n. Similar to Theorem 2 in Chen et al (2023b), ρcls 1 and ρcls 2 are strongly consistent and asymptotically normally distributed with √ T − 1( θcls…”

“…Unfortunately, a critical limitation occurs for the signed binomial thinning operator or its extensions: the counting sequence excludes the counts of −1, i.e., −1 is ignored. To fill this gap, we use the trinomial difference thinning operator (Chen et al, 2023b) and define a trinomial difference autoregressive (TDBZAR) model for bounded Z-valued time series, which is a terrific contribution because the incorporated trinomial difference thinning operator includes the counts of −1 (besides of 0 and 1), and the sum of its counting sequence follows a trinomial difference distribution, which makes the conditional expectation take a linear form.…”