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Empirical Likelihood Inference for Random

Coeﬃcient INAR(p) Process

Haixiang Zhang, Dehui Wang Fukang Zhu

Institute of Mathematics, Jilin University, Changchun 130012, China

Abstract: In this paper, we study the empirical likelihood method for the pth-order random

coeﬃcient integer-valued autoregressive process. In particular, the limiting distribution of the

log empirical likelihood ratio statistic is established and the conﬁdence regions for the

parameter of interest are derived. Also a simulation study is conducted for the evaluation of

the developed approach.

Key words: Models of count data; Random coeﬃcient; INAR process; Empirical likelihood;

Asymptotic distribution.

0 Introduction

The empirical likelihood (EL) method was ﬁrst proposed by Owen (1988) and further

studied by Owen (1990) among others. It is a useful tool for statistical inference. Conﬁdence

regions based on the EL method do not have predetermined symmetry so that they can better

correspond with the true shape of the underlying distribution. The EL method has been

successfully applied to many areas including linear regression models (Owen, 1991), generalized

linear models (Kolaczyk, 1994), general estimation equations (Qin and Lawless, 1994), semi-

parametric models (Qin and Wong, 1996), and dependent processes (Kitamura, 1997). Also

Cui and Chen (2003) considered its application to linear errors-in-variables models, Wang and

Jing (1999, 2003) to partially linear models, and You and Zhou (2006) to semi-parametric

Ä7‘8µ National Natural Science Foundation of China (No.10971081), Specialized Research Fund for the

Doctoral Program of Higher Education (No.20070183023), Program for New Century Excellent Talents in University,

Scientiﬁc Research Fund of Jilin University (No.200810024), and 985 Project of Jilin University.

Šö{0µ Ü°Œ£1984-)§I§Æ¬ïÄ)§Ì‡ïÄ••µžmS©Û"Ï&ŠöµŸ£1969-¤§I§Ç§Ì‡

ïÄ••µžmS©Û§x°Ž. Email: wangdh@jlu.edu.cn

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varying-coeﬃcient partially linear regression models. More references and techniques can be

found in the monograph Owen (2001).

Integer-valued time series data occur in many ﬁelds including insurance, medicine, queueing

systems, communications, reliability theory and meteorology. Examples of such data include

the data on the number of patients in a hospital at a speciﬁc time, the number of bases of DNA

sequences, the number of road accidents in a certain area in successive months, and the number

of machines waiting for maintenance. In recent years, a number of authors have studied the

problem of integer-valued time series analysis and most models have been proposed based on

the thinning operator “ ◦ ” due to Steutel and van Harn (1979). Let X be a non-negative

integer-valued random variable and α ∈ [0, 1). Then “ ◦ ” is deﬁned as

α ◦X =

i=1

where {B

} is an independent and identically distributed (iid) Bernoulli random sequence with

P (B

= 1) = 1 −P (B

= 0) = α that is independent of X. Some processes have been discussed

by McKenzie (1986, 1988), Al-Osh and Alzaid (1987), Alzaid and Al-Osh (1988) and Al-Osh

and Aly (1992), and Zheng et al. (2006, 2007) introduced random coeﬃcient integer-valued

autoregressive models.

In this paper, an EL method is developed for the pth-order random coeﬃcient integer-

valued autoregressive (RCINAR(p)) process. In Section 2, we review the deﬁnition of the

RCINAR(p) process and propose the empirical likelihood ratio (ELR) statistic. In Section 3,

the asymptotic distribution of the ELR statistic is established and the conﬁdence regions for

the parameter of interest are derived. Section 4 presents some simulation results.

1 Empirical likelihood for the RCINAR(p) process

The RCINAR(p) process {X

} satisﬁes an equation of the form:

i=1

(t)

◦ X

t−i

+ Z

, t ≥ 1, (1)

where {φ

(t)

} is an iid sequence with a cumulative distribution function P

on [0,1) and {Z

}

is an iid non-negative integer-valued sequence with a probability mass function f

> 0. Also

in the above, {φ

(t)

, 1 ≤ i ≤ p} and {Z

} are independent sequences. Let λ = E(Z

), φ

E(φ

(t)

), i = 1, 2, ··· , p, and we suppose

i=1

< 1.

Now we introduce the EL method. Let x

, ··· , x

be iid discrete-valued d-dimensional

random vectors with common cumulative distribution function F and mean µ. The EL function

L(F ) =

t=1

dF (x

) =

t=1

(2)

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