Mathematical Problems in Engineering

Volume 2016, Article ID 9505794, 8 pages

http://dx.doi.org/10.1155/2016/9505794

## Empirical Likelihood Inference for First-Order Random Coefficient Integer-Valued Autoregressive Processes

College of Mathematics, Jilin Normal University, Siping 136000, China

Received 21 June 2015; Accepted 4 November 2015

Academic Editor: Mustafa Tutar

Copyright © 2016 Zhiwen Zhao and Wei Yu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We apply the empirical likelihood method to estimate the variance of random coefficient in the first-order random coefficient integer-valued autoregressive (RCINAR(1)) processes. The empirical likelihood ratio statistic is derived and some asymptotic theory for it is presented. Furthermore, a simulation study is presented to demonstrate the performance of the proposed method.

#### 1. Introduction

Integer-valued time series data are fairly common in practice. Especially in economics and medicine, many interesting variables are integer-valued. In the last three decades, integer-valued time series have received increasing attention because of their wide applicability in many different areas, and there were many developments in the literature on it. See, for instance, Davis et al. [1] and MacDonald and Zucchini [2]. For count data, so far there are two main classes of time series models that have been developed in recent years: state-space models and thinning models. For state-space models, we refer to Fukasawa and Basawa [3]. Integer-valued autoregressive (INAR(1)) model was first defined by Steutel and Harn [4] through the “thinning” operator . Recall the definition of a “thinning” operator : where is an integer-valued random variable and and is an i.i.d. Bernoulli random sequence with that is independent of . Based on the “thinning” operator , the INAR(1) model is defined aswhere is a sequence of i.i.d. nonnegative integer-valued random variables.

Note that the parameter may be random and it may vary with time; Zheng et al. [5] introduced the following first-order random coefficient integer-valued autoregressive (RCINAR(1)) model:where is an independent identically distributed sequence with cumulative distribution function on with and ; is a sequence of i.i.d. nonnegative integer-valued random variables with and . Moreover, and are independent.

Zheng et al. [6] further generalized the above model to the -order cases. In recent several years, RCINAR model has been studied by many authors (see references in [7–10]). In this paper, we are concerned with estimating the variance of random coefficient in model (3). We propose an empirical log-likelihood ratio statistics for and derive its asymptotic distribution which is standard .

As a nonparametric statistical method, the empirical likelihood method was introduced by Owen [11–13]. The advantages of the empirical likelihood are now widely recognized. It has sampling properties similar to the bootstrap. Many advantages of the empirical likelihood over the normal approximation-based method have also been shown in the literature. These attractive properties have motivated various authors to extend empirical likelihood methodology to other situations. Now, the empirical likelihood methods have been widely applied to the statistical inference of the time series models (see [14–21]).

The remainder of this paper is organized as follows: In Section 2, we introduce the methodology and the main results. Simulation results are given in Section 3. Section 4 provides the proofs of the main results.

Throughout the paper, we use the notations “” and “” to denote convergence in distribution and convergence in probability, respectively. Convergence “almost surely” is written as “a.s.” Furthermore, denotes the transpose matrix of the matrix , and denotes Euclidean norm of the matrix or vector.

#### 2. Methodology and Main Results

In this section, we will first discuss how to apply the empirical likelihood method to estimate the unknown parameter .

Let , and . For simplicity of notation, we write as ; parameters and will be omitted. Then, after simple algebra, we get and , where .

First we consider estimating by using the conditional least-squares method. Based on the sample , the least-squares estimator of can be obtained by minimizing with . Solving the equationfor , we haveLet , where and are given by Zheng et al. [5]. Further let and . Then, the estimating equation of can be written aswhere .

In what follows, we apply Owen’s empirical likelihood method to make inference about . For convenience of writing, let be a probability vector with and ; also, let denote the true parameter value for . The log empirical likelihood ratio evaluated at , a candidate value of , is By using the Lagrange multiplier method, introducing a Lagrange multiplier , we havewhere satisfies

Owen’s empirical log-likelihood ratio statistic has a chi-squared limiting distribution. Similarly, we can prove that will also be asymptotically chi-squared distributed. In order to establish a theory for , we assume that the following assumptions hold:() is a strictly stationary and ergodic process.().

*Remark 1. *Similar conditions can be found in [8].

Now we can give the limiting properties of .

Theorem 2. *Assume that and hold. If is the true value of , thenwhere is a chi-squared distribution with degree of freedom.*

As a consequence of the theorem, confidence regions for the parameter can be constructed by (12). For , an asymptotic confidence region for is given by where is the upper -quantile of the chi-squared distribution with degrees of freedom equal to 1.

#### 3. Simulation Study

In this section, we conduct some simulation studies which show that our proposed methods perform very well.

In the first simulation study, we consider the RCINAR(1) process:where is a sequence of i.i.d. sequence with and ; . We take and and take and . Samples of size and All simulation studies are based on 1000 repetitions. The results of the simulations are presented in Table 1. The nominal confidence level is chosen to be and , and the figures in parentheses are the simulation results at the nominal level of .