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 .
From Table 1, we find that the confidence region obtained by using the empirical likelihood method has high coverage levels for different . The coverage probability has no obvious change for different and . 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.
4. Proofs of the Main Results
Lemma 3. Assume that and hold. Thenwhere and .
Proof. Note thatFirst, we consider . After simple algebra calculation, we haveNext, we consider . By the mean value theorem, we have where lies between and and lies between and . Therefore,Below, we prove that . For , note that By Theorem 3.1 in Zheng et al. [5], we know thatMoreover, by the ergodic theorem, we haveFurther note thatwhich, combined with (22) and (23), implies thatSimilarly, we can prove thatNext, we prove thatNote thatBy the ergodic theorem, we haveBy (21), we haveTherefore, by (21), we haveSimilarly, we can prove thatUsing this, together with (25), (26), and (27), we can proveFinally, we prove thatFor this, we first prove thatBy the Cramer-Wold device, it suffices to show that, for all ,Let and Then is a zero-mean, square integrable martingale array. By making use of a martingale central limit theorem [22], 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 and hold. Then
Proof. Note thatBy (23), in order to prove Lemma 4, we have only to show thatNote thatBy the ergodic theorem, we know thatSimilar to the proof of (33), we can further prove thatThis, in conjunction with (41), yields (39). So we complete the proof of Lemma 4.
Lemma 5. Assume that and hold. Then
Proof. To prove (43), we only need to prove thatLet For , definewhere denotes the largest integer not greater than . For each , (37) implies that as . Moreover, note that For given , choose so that Therefore, for each , if , then we have So, for any ,which implies (44). So we prove (43).
Proof of Theorem 2. First, we prove thatWrite , where and Observe that This implies thatFurther, by Lemma 4, we know that By Lemma 3, we have Thus by (51) and Lemma 5, we have which implies (49).
By (49) and Lemma 5, we can prove thatExpanding (11), we haveBy (55) and Lemmas 3, 4, and 5, we know that the final term in (56) is bounded byThis, together with (41), yieldsBy the Taylor expansion, we have Below, we prove that there exists a finite number , such thatThe Taylor expansion of around yields where, as . Thus, there exists , such that for any Moreover, by (55), we have Let . Note that if , then for ,which implies that where
Moreover, by (10) and (58), we haveThis, 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).