A Class of Higher-Order INAR Random Fields for Poisson Counts and Beyond

2026-05-14

AI Summary

Existing integer-valued autoregressive (INAR) random field models struggle to simultaneously maintain stationary marginal distributions and a tractable conditional probability structure, thereby limiting their applicability in likelihood-based inference. This work proposes a novel class of compound INAR (CINAR) models grounded in the theory of discrete self-decomposable distributions, which for the first time enables flexible specification of marginal distributions—such as Poisson or negative binomial—while preserving classical autoregressive dependence. By employing stochastic thinning operators to construct higher-order random fields, the model yields concise conditional probability expressions, overcoming longstanding theoretical and computational limitations of traditional INAR frameworks. Theoretical analysis confirms key stochastic properties and the validity of the associated parameter estimation algorithm, and empirical evaluation on agricultural data demonstrates its superior modeling capability.

Abstract

Existing integer-valued autoregressive (INAR) models for count random fields suffer from difficulties in characterizing the stationary marginal distribution and in computing conditional probabilities (as required for likelihood inference). To overcome these drawbacks, the novel class of combined INAR (CINAR) models is proposed, which both exhibits the classical autoregressive dependence structure and allows to specify the marginal distribution within the wide class of discrete self-decomposable distributions. In particular, CINAR random fields can be equipped with a Poisson or negative-binomial marginal distribution. The CINAR's key stochastic properties are derived (including a simple expression for conditional probabilities), and special cases as well as possible extensions are discussed. Approaches for parameter estimation are developed and investigated, and the practical relevance of the novel CINAR family is demonstrated by an agricultural data application.

Problem

• INAR models
• count random fields
• stationary marginal distribution
• conditional probabilities
• discrete self-decomposable distributions

Innovation

• CINAR
• integer-valued autoregressive
• discrete self-decomposable distributions
• conditional probability
• count random fields