Abstract
In this paper, the authors consider the application of the blockwise empirical likelihood method to the partially linear single-index model when the errors are negatively associated, which often exist in sequentially collected economic data. Thereafter, the blockwise empirical likelihood ratio statistic for the parameters of interest is proved to be asymptotically chi-squared. Hence, it can be directly used to construct confidence regions for the parameters of interest. A few simulation experiments are used to illustrate our proposed method.
1. Introduction
The partially linear single-index model is as follows:where is a response variable, is the covariate, is an unknown univariate measurable function, is a random error, and is an unknown vector in with (where denotes the Euclidean norm). The restriction assures identifiability.
Model (1) is flexible enough to include many important statistical models, so it has attracted much attention and has been extensively studied in recent years. Relevant studies about Model (1) have been done by [1–5]; all of which are based on the independent error sequence. In practice, all the parts of the random error sequence are often associated with each other, such as negatively associated errors, m-dependent errors, and ARCH errors, so that the abovementioned findings cannot be used directly. Therefore, it is necessary to study Model (1) with associated errors.
The finite random variable sequence is negatively associated (NA); if, on the condition of any two arbitrary disjoint subsets , and any real-valued coordinate-wise nondecreasing functions and , it is established that there is
As for infinite random variable sequence, if any arbitrary finite subset is negatively associated, the infinite sequence is negatively associated.
NA sequence has been introduced and studied by the authors of [6, 7] since the 1980s. Because the NA sequence includes the independent sequence, it has been widely applied in multivariate statistical analysis, the permeability analysis, and reliability theory drew much attention, and a lot of research results have been obtained. Under the fields of NA random variables, the author of [8] presented the asymptotic normality and central limit theorem; the authors of [9] proved the law of the iterated logarithm; the authors of [10] studied the exponential inequality, and so on.
However, there is little research about the partially linear single-index model under NA error. This paper, with the enlightenment of [11, 12], focuses on estimating , with blockwise empirical likelihood when the errors are subjected to NA in Model (1). Throughout this paper, we assume that the data are generated from Model (1), and are NA errors with and in Model (1).
The rest of this paper is organized as follows. In Section 2, the blockwise empirical likelihood method and the relative asymptotic result are presented. In Section 3, some simulations are conducted to illustrate the proposed approach. All proofs are shown in Section 4.
2. Methodology and Main Results
2.1. Bias-Corrected Blockwise Empirical Likelihood
In this part, we will use the bias-corrected blockwise empirical likelihood to construct the confidence region for . For this reason, first we introduce an auxiliary vector using the bias-corrected method of [2]. The details are as follows. Now that means that the value is a boundary point on the unit sphere and does not have a derivative at the point of the parameter space. Nevertheless, the derivative of on in the building of the empirical likelihood ratio statistics must be used. Thereby, we adopt the “delete-one-component” method which is widely used in the semiparameter model. Let and be a -dimensional parameter vector after deleting the rth component of . Without losing its generality, we assume . Then, we can write
Since can be determined by , only the confidence regions of need to be taken into consideration. Moreover, , which means that is infinitely differentiable in a neighborhood of the parameter . Thus, the Jacobian matrix iswhere is a -dimensional vector with th component 1 and .
Suppose and and the auxiliary vector is defined aswhere is the derivative of with respect to . Note that on the condition that is the true parameter. Therefore, the empirical likelihood by [13] is used to construct the bias-corrected empirical log-likelihood, which is defined as
Since , , , and are unknown, formula (5) cannot be used to construct the confidence regions directly. To replace them with their estimators is what the researchers usually do. Next, we apply the local linear smooth method of [14] to obtain the estimators of and . For any fixed , we focus on finding a and b to minimizewhere , is a kernel function and is a bandwidth. Let be the solution to minimize (7). Through a simple calculation,where
Then, we defineandas the estimators of and , respectively.
When (8), (9), (11), and (12) are plugged in (5) and (6), an estimated auxiliary vector and an estimated bias-corrected empirical log-likelihood ratio can be, respectively, defined as follows:
Under the independent identically distributed errors, the empirical likelihood ratio statistic is constructed by [2]; and its asymptotic result is presented. In this paper, may be dependent when the error satisfies NA, so the method of [2] cannot be applied directly. For this, we apply the small-block and large-block arguments to construct the blockwise empirical likelihood ratio. Let , where denotes the integral part of , and and are positive integers satisfying . For , letwhere
By using the Lagrange multiplier method, the bias-corrected blockwise empirical likelihood ratio statistic iswhere is determined by
2.2. Asymptotic Result
In this subsection, the main result of this paper is summarized. In order to state the asymptotic result, the following assumptions will be used:(i): the density function of is bounded away from 0 on and satisfies the Lipschitz condition of order 1 on , where and is a compact support set of (ii): , , and have two bounded and continuous derivatives on , where and are the sth and lth components of and , , respectively(iii): the kernel is a bounded symmetric density function and satisfies(iv); (v): the bandwidth satisfies that , , and (vi) and are both positively definite matrices, where (vii)(viii): when , (ix), and
Remark 1. According to [2], – guarantee the asymptotic distribution theory. is a common assumption in the NA situation. constrains the block size in order to obtain the desired results.
Theorem 1. Assume that – are satisfied. If is the true value of the parameter and , when , thenwhere stands for the convergence in distribution.
Based on Theorem 1, can be used to construct confidence regions for . For any given , there exists which makes tenable, and thenwhich is the confidence regions of with the asymptotically correct coverage probability .
3. Simulation
In this section, we use two examples to conduct some simulation studies to compare the performance of the proposed empirical likelihood method (ELM) and the normal approximation method (NAM).
We assume , and . Then, we can get for and for , otherwise . Thus, is a NA sequence, and its mean is approximately zero. According to , the rate of is between and . Therefore, the bandwidth is selected as by using the cross-validation (CV) method. The details of the CV method have been discussed in the references [15] of the study by Hrdle et al. [15], This selected bandwidth satisfies the condition . Let and , which satisfy the condition .
Example 1. In the simulation, we generate 1000 datasets, each consisting of . The set of data is generated from the following model:where , , , and is the bivariate with independent components. The kernel function is taken as the Quartic kernel if .
Since , we only consider the confidence region of the parameter . The coverage probabilities of the empirical likelihood confidence regions and the normal approximation confidence regions, with the normal level 0.95, are reported in Table 1. As is expected, the results fit our theory fairly well. The larger the sample size is, the closer the empirical coverage probability is to the nominal level. The proposed empirical likelihood method outperforms the normal approximation method.
Figure 1 plots the proposed empirical likelihood confidence region and the normal approximation confidence regions for based on the confidence level of 0.95 when the sample size is 300.
Example 2. In this simulation, the coverage probabilities and average lengths of confidence intervals are calculated by the proposed empirical likelihood method and the normal approximation method. Consider Model (1) with and , where , , and . is independent and all from the uniform , and the two components of are from the bivariate standard normal distribution. The kernel function is taken as the Epanechnikov kernel if .
Based on 500 simulation runs, the simulation results are reported in Table 2. From Table 2, the following results can be obtained. The coverage probabilities of the empirical likelihood method and the normal approximation method are in agreement with the nominal level of 0.90; the empirical likelihood method has slightly smaller interval lengths compared with the normal approximation method.
4. Proofs
In order to prove Theorem 1, we first give some lemmas. Throughout this section, for a concise and convenient representation, we use to denote any constant which may take a different values for each appearance, use and to denote the smallest and largest eigenvalues of A, respectively, and write .
Lemma 1. Let and be any two sequences, thenwhere is an arbitrary arrangement of order .
The proof of Lemma 1. Firstly, we assume . Via Abel inequality, it follows that
Secondly, we assume , and it also follows that
Combining (23) and (24), then we get
Lemma 2. Let be any random variables with for some constants and. Then,
Refer to the proof of Lemma 1 of [11].
Lemma 3. Let be NA random variables with and be a sequence of real constants. Then, there exists a constant which only depends on the given number s so that
The proof of Lemma 3 can be finished with the work by [16].
Lemma 4. Assume that – hold, for any integer , we can obtain, uniformly over ,where and . Here, are, respectively, the sth and lth component of , .
The proof of Lemma 4 is similar to the proof of Lemma 3 by [12]. So, the details are omitted here.
Lemma 5. Suppose that it satisfies –, is the true value of the parameter, and the rth component of , thenandwhere , and is defined in .
The proof of Lemma 5. It is easy to show thatwhere
In order to prove (29), we firstly need to show that
Let . For the central limit theorem of , we refer to Lemma 5 of [11]; then, we can obtain (33).
Secondly, we need to show that and . Here, we consider . Let denote the sth component of . By and and Lemma 1 and Lemma 4, we can get
The other formulas mentioned above can be proved by using Lemma 1, Lemma 3, and Lemma 4 and the similar methods of [2]. These formulas, together with (31) and (33), complete the proof of (29).
The proof of (30) uses a similar method to the proof of Lemma 5 by [12]. Here, we only give some key steps. For any with , we will show that
Then, we need to show thatwhere .
Similar to (31), we havewhere
As arguments in [11], we can get (36). Similar decomposition and proof can be used in (37) and (38). Thus, (30) is completely proved.
Lemma 6. Under –, it follows that
The proof of Lemma 6. Now thatwe only need to prove
Let denote the sth component of , and the notation of Lemma 5 is used.
We consider the later components of . By a simple calculation, we can obtain
Applying Lemma 2 with ,
This implies that since . Combining the Markov inequality, Lemma 4 and , for any ,
Hence, . can be dealt similarly. Then,
Moreover,can be similarly dealt with. Thus, (43) can be shown.
Lemma 7. Under conditions –, it follows that
The proof of Lemma 7. Let and . Denoting , then can be expressed by , where and .
From (17), we have
It follows that
By a straightforward calculation, we can get
As discussed in [12], we can obtain
Using Lemma 5, Lemma 6, and (53), we can have
Therefore, it is easy to obtain
The proof of Theorem 1. Let , and the notation of Lemma 7 is used. Using a Taylor expansion in (16), we havewhere
Thus,
By calculating directly from (17), we obtainwhere
Therefore, it follows that
It is easy to get
Combining (61) and (62), (58) can be rewritten as
Applying Lemma 5, we have
Thus, we complete the proof of Theorem 1.
Data Availability
Simulation data were obtained from Monte Carlo experiment.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors read and approved the final manuscript.
Acknowledgments
This work was supported by the Philosophy and Social Sciences Planning Project of Guangdong Province during the “13th Five-Year” Plan Period (nos. GD18CYJ08 and GD20CJY50), National Social Science Foundation of China (no. 18CTQ032), Guangdong Province Educational Science “Thirteenth Five-Year Plan” Project (no. 2019GXJK272), Guangdong Province Research Project (no. 2020WT030), Guangdong Provincial Department of Education Project (no. 2020WQNCX141), and Guangdong Polytechnic of Science and Technology Research Project (nos. XJPY2018006 and JG201918).