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

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 [omicron]. Recall the definition of a "thinning" operator [omicron]:

[phi] [omicron] X = [X.summation over (i=1)][B.sub.i], (1)

where X is an integer-valued random variable and [phi] [member of] [0, 1] and [B.sub.i] is an i.i.d. Bernoulli random sequence with P([B.sub.i] = 1) = [phi] that is independent of X. Based on the "thinning" operator [omicron], the INAR(1) model is defined as

[X.sub.t] = [phi] [omicron] [X.sub.t-1] + [Z.sub.t], t [greater than or equal to] 1, (2)

where {[Z.sub.t]} is a sequence of i.i.d. nonnegative integer-valued random variables.

Note that the parameter 0 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:

[X.sub.t] = [[phi].sub.t] [omicron] [X.sub.t-1] + [Z.sub.t], t [greater than or equal to] 1, (3)

where {[phi].sub.t]} is an independent identically distributed sequence with cumulative distribution function [p.sub.[phi]] on [0, 1) with E([[phi].sub.t]) = [phi] and Var([[phi].sub.t]) = [[sigma].sup.2.sub.[phi]]{[Z.sub.t]} is a sequence of i.i.d. nonnegative integer-valued random variables with E([Z.sub.t]) = [lambda] and Var([Z.sub.t]) = [[sigma].sup.2.sub.z]. Moreover, {[[phi].sub.t]} and {[Z.sub.t]} are independent.

Zheng et al. [6] 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 [7-10]). In this paper, we are concerned with estimating the variance [[sigma].sup.2.sub.[phi]] of random coefficient in model (3). We propose an empirical log-likelihood ratio statistics for [[sigma].sup.2.sub.[phi]] and derive its asymptotic distribution which is standard [chi square].

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 [mathematical expression not reproducible] to denote convergence in distribution and convergence in probability, respectively. Convergence "almost surely" is written as "a.s." Furthermore, [A.sup.[tau].sub.k x p] denotes the transpose matrix of the k x p matrix [B.sub.kxp], and [parallel] x [parallel] 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 [[sigma].sup.2.sub.[phi]].

Let [theta] = [([phi](1 - [phi]) - [[sigma].sup.2.sub.[phi]], [[sigma].sup.2.sub.Z]).sup.[tau]], [beta] = [([[sigma].sup.2.sub.[phi]], [[theta].sup.[tau]]).sup.[tau]] and [R.sub.t]([phi], [lambda]) = [X.sub.t]-E([X.sub.t] | [X.sub.t-1]). For simplicity of notation, we write [R.sub.t](([phi], [lambda]) as [R.sub.t]; parameters ([phi] and [lambda] will be omitted. Then, after simple algebra, we get E([X.sub.t] | [X.sub.t-1]) = [phi][X.sub.t-1] + [lambda] and E([R.sup.2.sub.t] | [X.sub.t-1]) = [Z.sup.[tau]].sub.t][beta], where [Z.sub.t] = [([X.sup.2.sub.t-1], [X.sub.t-1], 1).sup.[tau]].

First we consider estimating [beta] by using the conditional least-squares method. Based on the sample [X.sub.0], [X.sub.1], ... , [X.sub.n], the least-squares estimator [??] of [beta] can be obtained by minimizing

Q = [n.summation over (t=1)] ([R.sup.2.sub.t] - E [([R.sup.2.sub.t] | [X.sub.t-1])).sup.2] (4)

with [beta]. Solving the equation

[partial derivative]Q/[partial derivative][beta] = [n.summation over (t=1)] ([R.sup.2.sub.t] - E ([R.sup.2.sub.t] | [X.sub.t-1]))[Z.sub.t] (5)

for [beta], we have

[??] [([n.summation over (t=1)] [Z.sub.t] [Z.sup.[tau].sub.t]).sup.-1] [n.summation over (t=1)][R.sup.2.sub.t][Z.sub.t]. (6)

Let [mathematical expression not reproducible], where [mathematical expression not reproducible] are given by Zheng et al. [5]. Further let [mathematical expression not reproducible]. Then, the estimating equation of [theta] can be written as

[mathematical expression not reproducible], (7)

where T = [(1, 0, 0).sup.[tau]].

In what follows, we apply Owen's empirical likelihood method to make inference about [[sigma].sup.2.sub.[phi]]. For convenience of writing, let

[mathematical expression not reproducible]; (8)

p = [([p.sub.1], ..., [p.sub.n]).sup.[tau]] be a probability vector with [[summation].sup.n.sub.t=1] [p.sub.t] = 1 and [p.sub.t] [greater than or equal to] 0; also, let [[sigma].sup.2.sub.[phi]0] denote the true parameter value for [[sigma].sup.2.sub.[phi]]. The log empirical likelihood ratio evaluated at [[sigma].sup.2.sub.[phi]], a candidate value of [[sigma].sup.2.sub.[phi]]0, is

[mathematical expression not reproducible]. (9)

By using the Lagrange multiplier method, introducing a Lagrange multiplier [lambda] [member of] R, we have

l([[sigma].sup.2.sub.[phi]]) = 2[n.summation over (t=1)] log (1 + [[lambda].sup.[tau]][H.sub.t]([[sigma].sup.2.sub.[phi]])), (10)

where [lambda] satisfies

[1/n] [n.summation over (t=1)] [H.sub.t]([[sigma].sup.2.sub.[phi]])/1 + [[lambda].sup.[tau]][H.sub.t]([[sigma].sup.2.sub.[phi]]) = 0. (11)

Owen's empirical log-likelihood ratio statistic has a chi-squared limiting distribution. Similarly, we can prove that l([[sigma].sup.2.sub.[phi]]) will also be asymptotically chi-squared distributed. In order to establish a theory for l([[sigma].sup.2.sub.[phi]]), we assume that the following assumptions hold:

([A.sub.1]) {[X.sub.t]} is a strictly stationary and ergodic process.

([A.sub.2]) E[[absolute value of [X.sub.t]].sup.8] < [infinity].

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

Now we can give the limiting properties of l([[sigma].sup.2.sub.[phi]]).

Theorem 2. Assume that ([A.sub.1]) and ([A.sub.2]) hold. If [[sigma].sup.2.sub.[phi]0] is the true value of [[sigma].sup.2.sub.[phi]], then

[mathematical expression not reproducible], (12)

where [[chi square].sub.1] is a chi-squared distribution with 1 degree of freedom.

As a consequence of the theorem, confidence regions for the parameter [[sigma].sup.2.sub.[phi]] can be constructed by (12). For 0 < [delta] < 1, an asymptotic 100(1 - [delta])% confidence region for [[sigma].sup.2.sub.[phi]] is given by

[mathematical expression not reproducible], (13)

where [[chi square].sub.1]([delta]) is the upper [delta]-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:

[X.sub.t] = [[phi].sub.t] [omicron] [X.sub.t-1] + [Z.sub.t], t [greater than or equal to] 1, (14)

where {[phi].sub.t]} is a sequence of i.i.d. sequence with E([[phi].sub.t]) = [phi] and Var([phi].sub.t]) = [[sigma].sup.2.sub.[phi]]; [Z.sub.t] ~ Poisson([lambda]). We take [phi] = 0.1, 0.2, 0.3, 0.4 and 0.5 and take [lambda] = 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.

From Table 1, we find that the confidence region obtained by using the empirical likelihood method has high coverage levels for different [[sigma].sup.2.sub.[phi]]. The coverage probability has no obvious change for different [phi] and [lambda]. 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 ([A.sub.1]) and ([A.sub.2]) hold. Then

[mathematical expression not reproducible], (15)

where W = E[([Z.sub.t][Z.sup.[tau].sub.t]([R.sup.2.sub.t] - [Z.sup.[tau].sub.t][beta]).sup.2]) and [GAMMA] = E([Z.sub.t][Z.sup.[tau].sub.t]).

Proof. Note that

[mathematical expression not reproducible]. (16)

First, we consider [A.sub.n3]. After simple algebra calculation, we have

[mathematical expression not reproducible]. (17)

Next, we consider [A.sub.n1]. By the mean value theorem, we have

[mathematical expression not reproducible], (18)

where [[phi].sup.*] lies between [??] and [phi] and [[lambda].sup.*] lies between [??] and [lambda]. Therefore,

[mathematical expression not reproducible]. (19)

Below, we prove that [B.sub.ni] - [o.sub.p](1), i - 1, 2, 3, 4, 5, 6. For [B.sub.n1], note that

[B.sub.n1] = [T.sup.[tau]] [(1/n [n.summation over (t=1)] [Z.sub.t][Z.sup.[tau].sub.t]).sup.-1] -2/n x [n.summation over (t=1)][Z.sub.t][R.sub.t]([phi], [lambda]) [X.sub.t-1] ([square root of n]([??] - [phi])). (20)

By Theorem 3.1 in Zheng et al. [5], we know that

[square root of (n)] ([??] - [phi]) = [O.sub.p] (1). (21)

Moreover, by the ergodic theorem, we have

[mathematical expression not reproducible], (22)

[mathematical expression not reproducible]. (23)

Further note that

[mathematical expression not reproducible] (24)

which, combined with (22) and (23), implies that

[B.sub.n1] = [O.sub.p](1)[o.sub.p](1)[O.sub.p](1) = [o.sub.p](1). (25)

Similarly, we can prove that

[[beta].sub.n2] = [o.sub.p] (1). (26)

Next, we prove that

[B.sub.n3] = [o.sub.p](1). (27)

Note that

[mathematical expression not reproducible]. (28)

By the ergodic theorem, we have

[2/n] [n.summation over (t=1)][absolute value of ([Z.sub.t][X.sup.2.sub.t-1])] = [O.sub.p](1). (30)

By (21), we have

[[parallel][square root of (n)]([??] - [phi])[parallel].sup.2] = [O.sub.p](1). (30)

Therefore, by (21), we have

[B.sub.n3] = [O.sub.p](1) [o.sub.p] (1) [O.sub.p] (1) [O.sub.p] (1) = [o.sub.p] (1). (31)

Similarly, we can prove that

[B.sub.ni] = [o.sub.p] (1), i = 4, 5, 6. (32)

Using this, together with (25), (26), and (27), we can prove

[A.sub.n1] = [o.sub.p] (1). (33)

Finally, we prove that

[mathematical expression not reproducible]. (34)

For this, we first prove that

[mathematical expression not reproducible]. (35)

By the Cramer-Wold device, it suffices to show that, for all c [member of] [R.sup.3]\(0, 0, 0),

[mathematical expression not reproducible]. (36)

Let [[xi].sub.nt] = (1/[square root of (n)])[c.sup.[tau]][Z.sub.t]([R.sup.2.sub.t] - [Z.sup.[tau].sub.t][beta]) and [F.sub.nt] = [sigma]([[xi].sub.nr], 1 [less than or equal to] r [less than or equal to] t). Then {[[summation].sup.n.sub.t=1] [[xi].sub.nt], [F.sub.nt], 1 [less than or equal to] t [less than or equal to] n, n [greater than or equal to] 1} 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 ([A.sub.1]) and ([A.sub.2]) hold. Then

[mathematical expression not reproducible]. (37)

Proof. Note that

[mathematical expression not reproducible]. (38)

By (23), in order to prove Lemma 4, we have only to show that

[mathematical expression not reproducible]. (39)

Note that

[mathematical expression not reproducible]. (40)

By the ergodic theorem, we know that

[mathematical expression not reproducible]. (41)

Similar to the proof of (33), we can further prove that

[C.sub.ni] = [o.sub.p] (1), i = 2, 3, 4, 5, 6. (42)

This, in conjunction with (41), yields (39). So we complete the proof of Lemma 4.

Lemma 5. Assume that ([A.sub.1]) and ([A.sub.2]) hold. Then

[mathematical expression not reproducible]. (43)

Proof. To prove (43), we only need to prove that

[mathematical expression not reproducible]. (44)

Let [T.sup.[tau]][T.sup.-1]W[[GAMMA].sup.-1]T = [[sigma].sup.2]. For m [member of] {1, ..., n}, define

[mathematical expression not reproducible], (45)

where [n(j/m)] denotes the largest integer not greater than n(j/m). For each m, (37) implies that [mathematical expression not reproducible]. Moreover, note that

[mathematical expression not reproducible]. (46)

For given s [member of] [0, 1], choose j [member of] {1, ..., m} so that s [member of] [(j - 1)/ m, j/m]. Therefore, for each s [member [0, 1], if [omega] [member of] [B.sub.nm], then we have

[mathematical expression not reproducible]. (47)

So, for any m [greater than or equal to] 1,

[mathematical expression not reproducible], (48)

which implies (44). So we prove (43).

Proof of Theorem 2. First, we prove that

[lambda] = [O.sub.p] ([n.sup.-1/2]). (49)

Write [lambda] = [rho][upsilon], where [rho] [greater than or equal to] 0 and [absolute value of [??]] = 1. Observe that

[mathematical expression not reproducible]. (50)

This implies that

[mathematical expression not reproducible]. (51)

Further, by Lemma 4, we know that

[[??].sup.2] [1/n] [n.summation over (t=1)][H.sup.2.sub.t]([[sigma].sup.2.sub.[phi]]) = [O.sub.p](1). (52)

By Lemma 3, we have

[1/n] [absolute value of ([upsilon] [n.summation over (t=1)] [H.sub.t]([[sigma].sup.2.sub.[phi]]))] = [O.sub.p]([n.sup. -1/2]). (53)

Thus by (51) and Lemma 5, we have

[rho] = [absolute value of [lambda]] = [O.sub.p]([n.sup.1/2]), (54)

which implies (49).

By (49) and Lemma 5, we can prove that

[mathematical expression not reproducible]. (55)

Expanding (11), we have

[mathematical expression not reproducible]. (56)

By (55) and Lemmas 3, 4, and 5, we know that the final term in (56) is bounded by

[mathematical expression not reproducible]. (57)

This, together with (41), yields

[lambda] = [(1/n [n.summation over (t=1)][H.sup.2.sub.t]([[sigma].sup.2.sub.[phi]])).sup.-1] 1/n [n.summation over (t=1)] [H.sub.t]([[sigma].sup.2.sub.[phi]]) + [o.sub.p]([n.sup.-1/2]). (58)

By the Taylor expansion, we have

log (1 + [lambda][H.sub.t] ([[sigma].sup.2.sub.[phi]])) = [lambda][H.sub.t]([[sigma].sup.2.sub.[phi]]) - [([lambda][H.sub.t]([[sigma].sup.2.sub.[phi]])).sup.2]/2 + [[phi].sub.t]. (59)

Below, we prove that there exists a finite number Q > 0, such that

P([absolute value of [[phi].sub.t]] [less than or equal to] Q[[absolute value of [lambda][H.sub.t] ([[sigma].sup.2.sub.[phi]])].sup.3], 1 [less than or equal to] t [less than or equal to] n) [right arrow] 1 as n [right arrow] [infinity]. (60)

The Taylor expansion of log(1 + x) around x = 0 yields

log (1 + x) = x - [x.sup.2]/2 + [x.sup.3]/3 + [omega](x), (61)

where, as x [right arrow] 0, [omega](x)/[x.sup.3] [right arrow] 0. Thus, there exists [iota] > 0, such that [absolute value of [omega](x)/[x.sup.3]] < 1/6 for any [absolute value of x] < [iota]. Moreover, by (55), we have

[mathematical expression not reproducible]. (62)

Let [A.sub.n] = {[omega]: [max.sub.1 [less than or equal to] t [less than or equal to] n][[absolute value of [lambda][H.sub.t]([[sigma].sup.2.sub.[phi]])].sup.3] < [[iota].sup.3]}. Note that if [omega] [member of] [A.sub.n], then for 1 [less than or equal to] t [less than or equal to] n,

[mathematical expression not reproducible], (63)

which implies that

P([absolute value of [[phi].sub.t]] [less than or equal to] Q [[absolute value of [lambda][H.sub.t] ([[sigma].sup.2.sub.[phi]])].sup.3], 1 [less than or equal to] t [less than or equal to] n) [right arrow] 1 as n [right arrow] [infinity], (64)

where Q = 1/2.

Moreover, by (10) and (58), we have

[mathematical expression not reproducible]. (65)

This, together with Lemmas 3 and 4, implies Theorem 2.

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

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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Zhiwen Zhao and Wei Yu

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

Correspondence should be addressed to Zhiwen Zhao; zhaozhiwenjilin@126.com

Received 21 June 2015; Accepted 4 November 2015

Academic Editor: Mustafa Tutar
Table 1: Coverage probabilities of the confidence intervals
on [[sigma].sup.2.sub.[phi]].

               [phi]      n = 50          n = 100         n = 300

[lambda] = 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)

[lambda] = 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)

Table 2: 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

Table 3: 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
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Title Annotation:Research Article
Author:Zhao, Zhiwen; Yu, Wei
Publication:Mathematical Problems in Engineering
Date:Jan 1, 2016
Words:4137
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