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On the novel geometric and negative binomial INAR(1) processes

Nadjib Bouzar

Abstract

Guerrero et al. \cite{GBSO} propose a novel approach to building first-order integer-valued autoregressive (\inar1) models based on the concept of thinning. The standard approach requires that the thinning operator be defined first and \inar1 models with either a specified marginal (the forward approach) or a specified innovation (the backward approach) are developed. In contrast, the approach in \cite{GBSO} is to start out by specifying both the marginal distribution of the process and that of its innovation sequence, and then proceed to identify the thinning operator by solving a functional equation. In this article we discuss the connection between the thinning operators the authors obtained for their novel geometric and negative binomial \inar1 models and the thinning operator introduced in \cite{AB1} and \cite{AB2}. More specifically, we show that the existence of the two models has been established in \cite{AB1} using the forward approach and a different parameterization. In the process, we strenghthen some of the authors' results obtained for the novel geometric \inar1 process and we extend their results to the novel negative binomial \inar1 process.

On the novel geometric and negative binomial INAR(1) processes

Abstract

Guerrero et al. \cite{GBSO} propose a novel approach to building first-order integer-valued autoregressive (\inar1) models based on the concept of thinning. The standard approach requires that the thinning operator be defined first and \inar1 models with either a specified marginal (the forward approach) or a specified innovation (the backward approach) are developed. In contrast, the approach in \cite{GBSO} is to start out by specifying both the marginal distribution of the process and that of its innovation sequence, and then proceed to identify the thinning operator by solving a functional equation. In this article we discuss the connection between the thinning operators the authors obtained for their novel geometric and negative binomial \inar1 models and the thinning operator introduced in \cite{AB1} and \cite{AB2}. More specifically, we show that the existence of the two models has been established in \cite{AB1} using the forward approach and a different parameterization. In the process, we strenghthen some of the authors' results obtained for the novel geometric \inar1 process and we extend their results to the novel negative binomial \inar1 process.
Paper Structure (9 sections, 15 theorems, 76 equations)

This paper contains 9 sections, 15 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Some basic facts
  3. On the novel geometric INAR ($1$) process
  4. On the novel negative binomial INAR ($1$) process
  5. Estimation
  6. The conditional least squares method for the means
  7. The Yule-Walker estimation method for the means
  8. The conditional maximum likehood (CML) method for the means
  9. The conditional least squares method for the variances

Key Result

Lemma 3.1

Assume that $X$ is a $\mathbb{Z}_+$-valued random variables. Then, the conditional probability of $(\alpha,\mu) \star X$ given $X=x$ is where, for $n\ge 1$, $0\le i\le n$, $1\le l\le n$, and $0<y<1$,

Theorems & Definitions (24)

  • Lemma 3.1
  • proof
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • Proposition 3.2
  • proof
  • Lemma 4.1
  • Proposition 4.1
  • Proposition 4.2
  • ...and 14 more