Fractional approaches for the distribution of innovation sequence of INAR(1) processes

TL;DR

This paper introduces four fractional methods to derive the innovation distribution in INAR(1) processes, revealing geometric-type distributions and proposing new models for overdispersed count data.

Contribution

It develops four novel fractional approaches for the innovation distribution in INAR(1) models and introduces four new autoregressive models with known marginals.

Findings

Distribution of innovations has geometric-type distribution.

Four new autoregressive models for overdispersed counts.

Methods applicable for autocorrelated count data analysis.

Abstract

In this paper, we present a fractional decomposition of the probability generating function of the innovation process of the first-order non-negative integer-valued autoregressive [INAR(1)] process to obtain the corresponding probability mass function. We also provide a comprehensive review of integer-valued time series models, based on the concept of thinning operators, with geometric-type marginals. In particular, we develop four fractional approaches to obtain the distribution of innovation processes of the INAR(1) model and show that the distribution of the innovations sequence has geometric-type distribution. These approaches are discussed in detail and illustrated through a few examples. Finally, using the methods presented here, we develop four new first-order non-negative integer-valued autoregressive process for autocorrelated counts with overdispersion with known marginals, and derive some properties of these models.