A simple integer-valued bilinear time series model

Abstract

In this paper, we extend the integer-valued model class to give a nonnegative integer-valued bilinear process, denoted by INBL(p,q,m,n), similar to the real-valued bilinear model. We demonstrate the existence of this strictly stationary process and give an existence condition for it. The estimation problem is discussed in the context of a particular simple case. The method of moments is applied and the asymptotic joint distribution of the estimators is given: it turns out to be a normal distribution. We present numerical examples and applications of the model to real time series data on meningitis and Escherichia coli infections.

Note on integer-valued bilinear time series models

“…Theorem 2.2 provides sufficient parameter restrictions ensuring the existence of (higher-order) moments under the stationary distribution. Along completely different lines our results generalize Sections 2 and 3 in Doukhan et al (2006) who restricted attention to the INBL(1, 0, 1, 1) process. Moreover our parameter restrictions in Theorem 2.2 are less severe.…”

“…This note reconsiders the nonnegative integer-valued bilinear processes introduced by Doukhan, Latour, and Oraichi (2006). Using a hidden Markov argument, we extend their result of the existence of a stationary solution for the INBL(1,0,1,1) process to the class of superdiagonal INBL(p, q, m, n) models.…”

“…The last two decades there were many developments in the literature on integer-valued time series; see McKenzie (2003) for a detailed review; for example, INteger-valued Moving Average processes (Al-Osh and Alzaid (1991)), INteger-valued AutoRegressive processes (Al-Osh and Alzaid (1987), Al-Osh and Alzaid (1990), and Du and Li (1991)), and Generalized INteger-valued AutoRegressive processes (Latour (1998)) are common choices. Doukhan et al (2006) introduced the class of nonnegative INteger-valued BiLinear time series, which contains the three aforementioned classes. Higher order processes turn out to be highly releveant see (Latour (1998)) for the class of GINAR models.…”

“…Moreover our parameter restrictions in Theorem 2.2 are less severe. Estimation of superdiagonal INBL processes can be performed along the same lines as in Doukhan et al (2006); The details are beyond the scope of this note.…”

“…Using a hidden Markov argument, we extend their result of the existence of a stationary solution for the INBL(1,0,1,1) process to the class of superdiagonal INBL(p, q, m, n) models. Our approach also yields improved parameter restrictions for several moment conditions compared to the ones in Doukhan, Latour, and Oraichi (2006). …”