Integer-valued autoregressive processes with prespecified marginal and innovation distributions: a novel perspective

Abstract

Integer-valued autoregressive (INAR) processes are generally defined by specifying the thinning operator and either the innovations or the marginal distributions. The major limitations of such processes include difficulties in deriving the marginal properties and justifying the choice of the thinning operator. To overcome these drawbacks, we propose a novel approach for building an INAR model that offers the flexibility to prespecify both marginal and innovation distributions. Thus, the thinning operator is no longer subjectively selected but is rather a direct consequence of the marginal and innovation distributions specified by the modeler. Novel INAR processes are introduced following this perspective; these processes include a model with geometric marginal and innovation distributions (Geo-INAR) and models with bounded innovations. We explore the Geo-INAR model, which is a natural alternative to the classical Poisson INAR model. The Geo-INAR process has interesting stochastic properties, such as MA($\infty$) representation, time reversibility, and closed forms for the $h^{\text{th}}$-order transition probabilities, which enables a natural framework to perform coherent forecasting. To demonstrate the real-world application of the Geo-INAR model, we analyze a count time series of criminal records in sex offenses using the proposed methodology and compare it with existing INAR and integer-valued generalized autoregressive conditional heteroscedastic models.

Keywords:

• Count time series
• geometric process
• thinning operator
• coherent forecasting
• time reversibility

Acknowledgments

We thank the Associate Editor and the anonymous referees for their careful review and insightful suggestions that helped improve the article.

Notes

1 Not to be confused with our novel geometric INAR(1) process (Geo-INAR(1), in short), which will be introduced in Subsection 2.1.

2 For a thinning operator “$°$” we have that $\alpha °X\le X.$ For an expectation-thinning operator “$\odot ,$” we mean that this is not necessarily true, but that $E(\alpha \odot X|X)\le X$ holds; see Aly and Bouzar^[3]. In this article, we use the term thinning operator indiscriminately.

3 Note that replacing r = 1 in Eq. (5) leads to $P(X_t=x)=\mu /{(1+\mu )}^{x+1},$ which is a different parameterization with respect to the geometric distribution considered in Subsection 2.1.

Additional information

Funding

We would also like to acknowledge support from the KAUST Research Fund (Grant No.: NIH 1R01EB028753-01). Part of this study was performed by Matheus B. Guerrero (Master’s Thesis) at the Department of Statistics of the Universidade Federal de Minas Gerais. W. Barreto-Souza also thanks Conselho Nacional de Desenvolvimento Científico e Tecnológico for financial support (CNPq-Brazil; grant number: 305543/2018-0).