MPE Mathematical Problems in Engineering 1563-5147 1024-123X Hindawi Publishing Corporation 10.1155/2016/9505794 9505794 Research Article Empirical Likelihood Inference for First-Order Random Coefficient Integer-Valued Autoregressive Processes Zhao Zhiwen 1 Yu Wei 1 Tutar Mustafa College of Mathematics Jilin Normal University Siping 136000 China jlnu.edu.cn 2016 822016 2016 21 06 2015 04 11 2015 822016 2016 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.

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.  and MacDonald and Zucchini . 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 . Integer-valued autoregressive (INAR(1)) model was first defined by Steutel and Harn  through the “thinning” operator . Recall the definition of a “thinning” operator : (1)ϕX=i=1XBi,where X is an integer-valued random variable and ϕ[0,1] and Bi is an i.i.d. Bernoulli random sequence with P(Bi=1)=ϕ that is independent of X. Based on the “thinning” operator , the INAR(1) model is defined as(2)Xt=ϕXt-1+Zt,t1,where {Zt} is a sequence of i.i.d. nonnegative integer-valued random variables.

Note that the parameter ϕ may be random and it may vary with time; Zheng et al.  introduced the following first-order random coefficient integer-valued autoregressive (RCINAR(1)) model:(3)Xt=ϕtXt-1+Zt,t1,where {ϕt} is an independent identically distributed sequence with cumulative distribution function pϕ on [0,1) with E(ϕt)=ϕ and Var(ϕt)=σϕ2; {Zt} is a sequence of i.i.d. nonnegative integer-valued random variables with E(Zt)=λ and Var(Zt)=σZ2. Moreover, {ϕt} and {Zt} are independent.

Zheng et al.  further generalized the above model to the p-order cases. In recent several years, RCINAR model has been studied by many authors (see references in ). In this paper, we are concerned with estimating the variance σϕ2 of random coefficient in model (3). We propose an empirical log-likelihood ratio statistics for σϕ2 and derive its asymptotic distribution which is standard χ2.

As a nonparametric statistical method, the empirical likelihood method was introduced by Owen . 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 ).

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 “d” and “p” to denote convergence in distribution and convergence in probability, respectively. Convergence “almost surely” is written as “a.s.” Furthermore, Ak×pτ denotes the transpose matrix of the k×p matrix Bk×p, 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 σϕ2.

Let θ=(ϕ(1-ϕ)-σϕ2,σZ2)τ, β=(σϕ2,θτ)τ and Rt(ϕ,λ)=Xt-E(XtXt-1). For simplicity of notation, we write Rt(ϕ,λ) as Rt; parameters ϕ and λ will be omitted. Then, after simple algebra, we get E(XtXt-1)=ϕXt-1+λ and E(Rt2Xt-1)=Ztτβ, where Zt=(Xt-12,Xt-1,1)τ.

First we consider estimating β by using the conditional least-squares method. Based on the sample X0,X1,,Xn, the least-squares estimator β^ of β can be obtained by minimizing (4)Q=t=1nRt2-ERt2Xt-12with β. Solving the equation(5)Qβ=t=1nRt2-ERt2Xt-1Ztfor β, we have(6)β^=t=1nZtZtτ-1t=1nRt2Zt.Let β~=β^(ϕ^,λ^), where ϕ^ and λ^ are given by Zheng et al. . Further let θ~=(0,1,1)β~ and β=(σϕ2,θ~τ)τ. Then, the estimating equation of θ can be written as(7)t=1nTτ1nt=1nZtZtτ-1ZtRt2ϕ^,λ^-Ztτβ=0,where T=(1,0,0)τ.

In what follows, we apply Owen’s empirical likelihood method to make inference about σϕ2. For convenience of writing, let (8)Htσϕ2=Tτ1nt=1nZtZtτ-1ZtRt2ϕ^,λ^-Ztτβ;p=(p1,,pn)τ be a probability vector with t=1npt=1 and pt0; also, let σϕ02 denote the true parameter value for σϕ2. The log empirical likelihood ratio evaluated at σϕ2, a candidate value of σϕ02, is (9)lσϕ2=-2maxt=1nptHtσϕ2=0t=1nlognpt. By using the Lagrange multiplier method, introducing a Lagrange multiplier λR, we have(10)lσϕ2=2t=1nlog1+λτHtσϕ2,where λ satisfies(11)1nt=1nHtσϕ21+λτHtσϕ2=0.

Owen’s empirical log-likelihood ratio statistic has a chi-squared limiting distribution. Similarly, we can prove that l(σϕ2) will also be asymptotically chi-squared distributed. In order to establish a theory for l(σϕ2), we assume that the following assumptions hold:

{Xt} is a strictly stationary and ergodic process.

E|Xt|8<.

Remark 1.

Similar conditions can be found in .

Now we can give the limiting properties of l(σϕ2).

Theorem 2.

Assume that (A1) and (A2) hold. If σϕ02 is the true value of σϕ2, then(12)lσϕ02dχ12asn,where χ12 is a chi-squared distribution with 1 degree of freedom.

As a consequence of the theorem, confidence regions for the parameter σϕ2 can be constructed by (12). For 0<δ<1, an asymptotic 100(1-δ)% confidence region for σϕ2 is given by (13)Cσϕ2δ=σϕ2R:lσϕ2χ12δ,where χ12(δ) is the upper δ-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:(14)Xt=ϕtXt-1+Zt,t1,where {ϕt} is a sequence of i.i.d. sequence with E(ϕt)=ϕ and Var(ϕt)=σϕ2; Zt~Poisson(λ). We take ϕ=0.1,0.2,0.3,0.4 and 0.5 and take λ=1 and 2. Samples of size n=50,100 and 300. 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 0.90 and 0.95, and the figures in parentheses are the simulation results at the nominal level of 0.90.

Coverage probabilities of the confidence intervals on σϕ2.

ϕ n = 50 n = 100 n = 300
λ = 1 0.1 0.980 (0.960) 0.976 (0.949) 0.979 (0.956)
0.2 0.972 (0.946) 0.984 (0.971) 0.984 (0.966)
0.3 0.977 (0.958) 0.986 (0.959) 0.990 (0.970)
0.4 0.983 (0.943) 0.984 (0.958) 0.984 (0.968)
0.5 0.978 (0.963) 0.986 (0.966) 0.990 (0.977)

λ = 2 0.1 0.980 (0.952) 0.972 (0.943) 0.970 (0.975)
0.2 0.977 (0.956) 0.980 (0.980) 0.973 (0.979)
0.3 0.989 (0.961) 0.983 (0.967) 0.969 (0.978)
0.4 0.983 (0.969) 0.983 (0.966) 0.974 (0.981)
0.5 0.973 (0.968) 0.982 (0.965) 0.970 (0.971)

From Table 1, we find that the confidence region obtained by using the empirical likelihood method has high coverage levels for different σϕ2. The coverage probability has no obvious change for different ϕ and λ. That means that the empirical likelihood method is also robust.

In the second simulation study, we illustrate how our method can be applied to fit a set of data through a practical example. We apply model (3) to fit the number of large- and medium-sized civil Boeing 767 cargo planes over the period 1985–2013 in China. The data in Table 2 are provided by the National Bureau of Statistics of China (http://data.stats.gov.cn/easyquery.htm?cn=C01). The fitting procedure is as follows: Firstly, by using the data over the period 1985–2003, we obtain the estimator of the model parameter. Then, by using this model, we can obtain a fitting sequence over the period 2004–2013. Furthermore, in order to compare with the ordinary autoregressive (AR(1)) model, we also give the fitting results of the AR(1) model. Table 3 reports the fitting results. In Table 3, Number is the true value and RCINAR(1) and AR(1) are the fitting results obtained by the RCINAR(1) model and AR(1) model, respectively. For the simulation results of AR(1) model, we take the rounded integer values of the simulation results. From the simulation results, we can find that the RCINAR(1) model has more plausible fitting results than the AR(1) model.

The number of Boeing 767 cargo planes.

 Year 1985 1986 1987 1988 1989 1990 Number 2 2 4 5 6 6 Year 1991 1992 1993 1994 1995 1996 Number 6 10 12 16 17 17 Year 1997 1998 1999 2000 2001 2002 Number 17 15 16 16 17 18 Year 2003 2004 2005 2006 2007 2008 Number 22 27 27 29 22 22 Year 2009 2010 2011 2012 2013 Number 19 18 15 13 11

The fitting results.

 Year 2004 2005 2006 2007 2008 2009 Number 27 27 29 22 22 19 RCINAR(1) 23 22 22 21 19 18 AR(1) 22 22 23 23 23 25 Year 2010 2011 2012 2013 Number 18 15 13 11 RCINAR(1) 18 18 16 16 AR(1) 25 27 27 29
4. Proofs of the Main Results Lemma 3.

Assume that (A1) and (A2) hold. Then(15)1nt=1nHtσϕ2dN0,TτΓ-1WΓ-1T,where W=E(ZtZtτ(Rt2-Ztτβ)2) and Γ=E(ZtZtτ).

Proof.

Note that(16)1nt=1nHtσϕ2=Tτ1nt=1nZtZtτ-11nt=1nZtRt2ϕ^,λ^-Ztτβ=Tτ1nt=1nZtZtτ-11nt=1nZtRt2ϕ^,λ^-Rt2ϕ,λ+Tτ1nt=1nZtZtτ-11nt=1nZtRt2ϕ,λ-Ztτβ+Tτ1nt=1nZtZtτ-11nt=1nZtZtτβ-βAn1+An2+An3.

First, we consider An3. After simple algebra calculation, we have(17)An3=Tτ1nt=1nZtZtτ-11nt=1nZtZtτβ-β=Tτ1nt=1nZtZtτ-11nt=1nZtZtτnβ-β=nTτβ-β=0.

Next, we consider An1. By the mean value theorem, we have (18)Rt2ϕ^,λ^-Rt2ϕ,λ=-2Rtϕ,λXt-1ϕ^-ϕ+λ^-λ,where ϕ lies between ϕ^ and ϕ and λ lies between λ^ and λ. Therefore,(19)An1=Tτ1nt=1nZtZtτ-11nt=1nZtRt2ϕ^,λ^-Rt2ϕ,λ=Tτ1nt=1nZtZtτ-1-2nt=1nZtRtϕ,λXt-1ϕ^-ϕ+λ^-λ=Tτ1nt=1nZtZtτ-1-2nt=1nZtRtϕ,λ+Rtϕ,λ-Rtϕ,λXt-1ϕ^-ϕ+λ^-λ=Tτ1nt=1nZtZtτ-1-2nt=1nZtϕ-ϕXt-1+λ-λ+Rtϕ,λXt-1ϕ^-ϕ+λ^-λ=Tτ1nt=1nZtZtτ-1-2nt=1nZtRtϕ,λXt-1ϕ^-ϕ-Tτ1nt=1nZtZtτ-12nt=1nZtRtϕ,λλ^-λ-Tτ1nt=1nZtZtτ-12nt=1nZtϕ-ϕXt-12ϕ^-ϕ-Tτ1nt=1nZtZtτ-12nt=1nZtλ-λXt-1ϕ^-ϕ-Tτ1nt=1nZtZtτ-12nt=1nZtϕ-ϕXt-1λ^-λ-Tτ1nt=1nZtZtτ-12nt=1nZtλ-λλ^-λBn1+Bn2+Bn3+Bn4+Bn5+Bn6.

Below, we prove that Bni=op(1),i=1,2,3,4,5,6. For Bn1, note that (20)Bn1=Tτ1nt=1nZtZtτ-1-2nt=1nZtRtϕ,λXt-1nϕ^-ϕ.By Theorem 3.1 in Zheng et al. , we know that(21)nϕ^-ϕ=Op1.Moreover, by the ergodic theorem, we have(22)-2nt=1nZtRtϕ,λXt-1p-2EZtRtϕ,λXt-1,(23)1nt=1nZtZtτpΓ.Further note that(24)EZtRtϕ,λXt-1=EERtϕ,λZtXt-1Ft-1=EZtXt-1ERtϕ,λFt-1=0,which, combined with (22) and (23), implies that(25)Bn1=Op1op1Op1=op1.Similarly, we can prove that(26)Bn2=op1.

Next, we prove that(27)Bn3=op1.Note that(28)Bn31nt=1nZtZtτ2nt=1nZtXt-12ϕ-ϕϕ^-ϕ1nt=1nZtZtτ2nt=1nZtXt-12ϕ-ϕ21nt=1nZtZtτ1n2nt=1nZtXt-12nϕ^-ϕ2.By the ergodic theorem, we have(29)2nt=1nZtXt-12=Op1.By (21), we have(30)nϕ^-ϕ2=Op1.Therefore, by (21), we have(31)Bn3=Op1op1Op1Op1=op1.Similarly, we can prove that(32)Bni=op1,i=4,5,6.Using this, together with (25), (26), and (27), we can prove(33)An1=op1.Finally, we prove that(34)An2dN0,TτΓ-1WΓ-1T.For this, we first prove that(35)1nt=1nZtRt2-ZtτβdN0,W.By the Cramer-Wold device, it suffices to show that, for all cR3(0,0,0),(36)1nt=1ncτZtRt2-ZtτβdN0,cτWc.Let ξnt=(1/n)cτZt(Rt2-Ztτβ) and Fnt=σ(ξnr,1rt). Then {t=1nξnt,Fnt,1tn,n1} is a zero-mean, square integrable martingale array. By making use of a martingale central limit theorem , we can prove (36). Further, by (23), we know that (34) holds. Therefore, by (17), (33), and (34), we can prove Lemma 3.

Lemma 4.

Assume that (A1) and (A2) hold. Then(37)1nt=1nHt2σϕ2pTτΓ-1WΓ-1T.

Proof.

Note that(38)1nt=1nHt2σϕ2=Tτ1nt=1nZtZtτ-11nt=1nZtZtτRt2ϕ^,λ^-Ztτβ21nt=1nZtZtτ-1T.By (23), in order to prove Lemma 4, we have only to show that(39)1nt=1nZtZtτRt2ϕ^,λ^-Ztτβ2pW.Note that(40)1nt=1nZtZtτRt2ϕ^,λ^-Ztτβ2=1nt=1nZtZtτRt2ϕ^,λ^-Rt2ϕ,λ+Rt2ϕ,λ-Ztτβ+Ztτβ-Ztτβ2=1nt=1nZtZtτRt2ϕ,λ-Ztτβ2+1nt=1nZtZtτRt2ϕ^,λ^-Rt2ϕ,λ2+1nt=1nZtZtτZtτβ-Ztτβ2+2nt=1nZtZtτRt2ϕ^,λ^-Rt2ϕ,λRt2ϕ,λ-Ztτβ+2nt=1nZtZtτRt2ϕ^,λ^-Rt2ϕ,λZtτβ-Ztτβ+2nt=1nZtZtτRt2ϕ,λ-ZtτβZtτβ-ZtτβCn1+Cn2+Cn3+Cn4+Cn5+Cn6.By the ergodic theorem, we know that(41)Cn1pW.Similar to the proof of (33), we can further prove that(42)Cni=op1,i=2,3,4,5,6.This, in conjunction with (41), yields (39). So we complete the proof of Lemma 4.

Lemma 5.

Assume that (A1) and (A2) hold. Then(43)max1tnHtσϕ2=opn1/2.

Proof.

To prove (43), we only need to prove that(44)1nmax1tnHt2σϕ2p0.Let TτΓ-1WΓ-1T=σ2. For m{1,,n}, define(45)Bn,mj=1nω:1nt=1nj/mHtσϕ2-jmσ21m,where [n(j/m)] denotes the largest integer not greater than n(j/m). For each m, (37) implies that P(Bn,m)P1 as n. Moreover, note that (46)max1tnHtσϕ2n1nsups0,1t=ns+1ns+1/mHtσϕ2. For given s[0,1], choose j{1,,m} so that s[(j-1)/m,j/m]. Therefore, for each s[0,1], if ωBn,m, then we have (47)1nt=ns+1ns+1/mHtσϕ21nt=nj-1/m+1nj+1/mHtσϕ2=1nt=1nj+1/mHtσϕ2-j+1mσ2-1nt=1nj-1/mHtσϕ2-j-1mσ2+2mσ22m+2mσ2=2m1+σ2. So, for any m1,(48)limnPmax1tnHtσϕ2n2m1+σ2limnPBn,m=1,which implies (44). So we prove (43).

Proof of Theorem <xref ref-type="statement" rid="thm2.1">2</xref>.

First, we prove that(49)λ=Opn-1/2.Write λ=ρϑ, where ρ0 and |ϑ|=1. Observe that (50)0=1nt=1nHtσϕ21+λHtσϕ2=ϑ1nt=1nHtσϕ21+λHtσϕ2=1nϑt=1nHtσϕ2-ρt=1nϑHt2σϕ21+ρϑHtσϕ2ρnϑ2t=1nHt2σϕ21+ρϑHtσϕ2-1nϑt=1nHtσϕ2ρnϑ2t=1nHt2σϕ21+ρmax1tnHtσϕ2-1nϑt=1nHtσϕ211+ρmax1tnHtσϕ2ρϑ21nt=1nHt2σϕ2-1nϑt=1nHtσϕ2.This implies that(51)11+ρmax1tnHtσϕ2ρϑ21nt=1nHt2σϕ21nϑt=1nHtσϕ2.Further, by Lemma 4, we know that (52)ϑ21nt=1nHt2σϕ2=Op1.By Lemma 3, we have (53)1nϑt=1nHtσϕ2=Opn-1/2.Thus by (51) and Lemma 5, we have (54)ρ=λ=Opn-1/2,which implies (49).

By (49) and Lemma 5, we can prove that(55)max1tnλHtσϕ2=Opn-1/2opn1/2=op1.Expanding (11), we have(56)0=1nt=1nHtσϕ21+λHtσϕ2=1nt=1nHtσϕ2-1nλt=1nHt2σϕ2+1nt=1nλ2Ht3σϕ21+λHtσϕ2.By (55) and Lemmas 3, 4, and 5, we know that the final term in (56) is bounded by(57)1nt=1nHtσϕ23λ21+λHtσϕ2-1=opn1/2Opn-1Op1=opn-1/2.This, together with (41), yields(58)λ=1nt=1nHt2σϕ2-11nt=1nHtσϕ2+opn-1/2.By the Taylor expansion, we have (59)log1+λHtσϕ2=λHtσϕ2-λHtσϕ222+φt.Below, we prove that there exists a finite number Q>0, such that(60)PφtQλHtσϕ23,1tn1asn.The Taylor expansion of log(1+x) around x=0 yields (61)log1+x=x-x22+x33+ϖx,where, as x0,ϖ(x)/x30. Thus, there exists ι>0, such that ϖ(x)/x3<1/6 for any |x|<ι. Moreover, by (55), we have (62)limnPmax1tnλHtσϕ23<ι3=1.Let An={ω:max1tn|λHt(σϕ2)|3<ι3}. Note that if ωAn, then for 1tn,(63)φtλHtσϕ23=λHtσϕ23/3+ϖλHtσϕ2λHtσϕ2313+16=12,which implies that (64)PφtQλHtσϕ23,1tn1asn,where Q=1/2.

Moreover, by (10) and (58), we have(65)lσϕ2=1nt=1nHtσϕ21nt=1nHt2σϕ2-11nt=1nHtσϕ2+op1.This, together with Lemmas 3 and 4, implies Theorem 2.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors acknowledge the financial supports by National Natural Science Foundation of China (nos. 11571138, 11271155, 11001105, 11071126, 10926156, and 11071269), Specialized Research Fund for the Doctoral Program of Higher Education (no. 20110061110003), Program for New Century Excellent Talents in University (NCET-08-237), Scientific Research Fund of Jilin University (no. 201100011), and Jilin Province Natural Science Foundation (nos. 20130101066JC, 20130522102JH, and 20101596).

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