Analogues are proposed for the concepts of self-decomposability and stability for distributions on the nonnegative integers. It turns out that these "discrete self-decomposable" and "discrete stable" distributions have properties that are quite similar to those of their continuous counterparts.

Discrete analogues of self-decomposability and stability

A Poisson distribution is commonly used as the innovation distribution for integer-valued autoregressive models, but its mean is equal to its variance, which limits flexibility, so a flexible, one-parameter, infinitely divisible Bell distribution may be a good alternative. In addition, for a parameter with a small value, the Bell distribution approaches the Poisson distribution. In this paper, we introduce a new first-order, non-negative, integer-valued autoregressive model with Bell innovations based on the binomial thinning operator. Compared with other models, the new model is not only simple but also particularly suitable for time series of counts exhibiting overdispersion. Some properties of the model are established here, such as the mean, variance, joint distribution functions, and multi-step-ahead conditional measures. Conditional least squares, Yule–Walker, and conditional maximum likelihood are used for estimating the parameters. Some simulation results are presented to access these estimates’ performances. Real data examples are provided.

https://www.mdpi.com/1099-4300/23/6/713/pdf

A New First-Order Integer-Valued Autoregressive Model with Bell Innovations.

This paper introduces a new stochastic process with values in the set Z of integers with sign. The increments of process are Poisson differences and the dynamics has an autoregressive structure. We study the properties of the process and exploit the thinning representation to derive stationarity conditions and the stationary distribution of the process. We provide a Bayesian inference method and an efficient posterior approximation procedure based on Monte Carlo. Numerical illustrations on both simulated and real data show the effectiveness of the proposed inference.

Generalized Poisson Difference Autoregressive Processes

. A new integer-valued autoregressive process (INAR) with Generalised Lagrangian Katz (GLK) innovations is deﬁned. We show that our GLK-INAR process is stationary, discrete semi-self-decomposable, inﬁnite divisible, and provides a ﬂexible modeling framework for count data allowing for under- and overdispersion, asymmetry, and excess of kurtosis. A Bayesian inference framework and an eﬃcient posterior approximation procedure based on Markov Chain Monte Carlo are provided. The proposed model family is applied to a Google Trend dataset which proxies the public concern about climate change around the world. The empirical results provide new evidence of heterogeneity across countries and keywords in the persistence, uncertainty, and long-run public awareness level.

First-order integer-valued autoregressive processes with Generalized Katz innovations

Over the past few years, interest has increased in models defined on positive and negative integers. Several application areas lead to data that are differences between positive integers. Some important examples are price changes measured discretely in financial applications, pre- and posttreatment measurements of discrete outcomes in clinical trials, the difference in the number of goals in sports events, and differencing of count-valued time series. This review aims at bringing together a wide range of models that have appeared in the literature in recent decades. We provide an extensive review on discrete distributions defined for integer data and then consider univariate and multivariate time-series models, including the class of autoregressive models, stochastic processes, and ARCH-GARCH– (autoregressive conditionally heteroskedastic–generalized autoregressive conditionally heteroskedastic–) type models. Expected final online publication date for the Annual Review of Statistics and Its Application, Volume 10 is March 2023. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

https://www.annualreviews.org/doi/pdf/10.1146/annurev-statistics-032921-022516

Models for Integer Data

In this article, we propose a new integer-valued autoregressive process with generalized Poisson difference marginal distributions based on difference of two quasi-binomial thinning operators. This model is suitable for data sets on ℤ = {..., -2, -1, 0, 1, 2,...} and can be viewed as a generalization of the Poisson difference INAR(1) process. An advantage of the difference of two generalized Poisson random variables is it can have longer or shorter tails compared to the Poisson difference distribution. We present some basic properties of the process like mean, variance, skewness, and kurtosis, and conditional properties of the process are derived. Yule-Walker estimators are considered for the unknown parameters of the model and a Monte Carlo simulation is presented to study the performance of estimators. An application to a real data set is discussed to show the potential for practice of our model.

A skew integer-valued time-series process with generalized Poisson difference marginal distribution

In this paper, we introduce a first order integer-valued autoregressive process with Borel innovations based on the binomial thinning. This model is suitable to modeling zero truncated count time s...

Enai Taveira da Cunha

Papers

A skew integer-valued time-series process with generalized Poisson difference marginal distribution

On shifted integer-valued autoregressive model for count time series showing equidispersion, underdispersion or overdispersion