Empirical Likelihood-Based Inference in Regressive Model with Moment Restrictions

In this paper, we consider the parameter estimation problem of linear regression model when the auxiliary information can be denoted by moment restrictions. We use the weighted least squares method to estimate the model parameters and to obtain the weights based on the auxiliary information by using the empirical likelihood method. /e limiting distribution of the estimator is established, and the simulation studies are carried out to demonstrate the feasibility of our theoretical results.

Linear regression model is widely used in empirical work in economics, medicine, and many other disciplines due to its simple form. In this paper, we consider the parameter estimating problem of linear regression model when auxiliary information is available. We care about how to decrease the estimation bias by using the auxiliary information based on the empirical likelihood method. e customary design-based estimator does not make use of auxiliary population information at the estimation stage. Observe that auxiliary information can be used to increase the precision of estimators in sample surveys. erefore, Chambers and Dunstan [1] propose a simple method for estimating the population distribution functions which allows auxiliary population information to be directly incorporated into the estimation process. Rao et al. [2] obtain the ratio and difference estimators of a population distribution function under a general sampling design by using auxiliary population information. Chen and Qin [3] show that the empirical likelihood method can be naturally applied to finite population inference problems by making effective use of the auxiliary information. Zhang [4] employs the method of empirical likelihood to construct confidence intervals for M-functionals in the presence of auxiliary information. Zhong and Rao [5] show that making effective use of auxiliary population information can lead to efficient estimators for making inferences on finite population parameters. GarcíA and Cebrián [6] derive confidence intervals for medians in a finite population by using multiauxiliary information through a multivariate regression-type estimator of the population distribution function. By assuming auxiliary information on the unknown distribution of the data, Crudu and Porcu [7] introduce a weighted Z-estimator for moment condition models. Moreover, Tang and Leng [8] propose a weighted least squares estimation which incorporates auxiliary information to the efficiency of the estimator. Hellerstein and Imbens [9] analyze the estimation of coefficients in regression models under moment restrictions in which the moment restrictions are derived from auxiliary data. But the moment restriction does not include nuisance parameter. We further consider the parameter estimation problem of linear regression model when the auxiliary information can be denoted by moment restrictions which include nuisance parameter based on the empirical likelihood method. Specifically, the auxiliary information can be expressed as moment constraint E(g(X i , θ 0 )) � 0, where g(X i , θ) � (g 1 (X i , θ), . . . , g r (X i , θ)), θ ∈ R d and r ≥ d. It is worth mentioning that θ can be different from α. erefore, moment constraint E(g(X i , θ 0 )) � 0 can denote a broad class of information. We will use the least squares method to estimate the model parameter and apply empirical likelihood method to obtain the weights by using the auxiliary information.
As a nonparametric method, empirical likelihood method is widely used in statistical inference of various models. Empirical likelihood method is first proposed by Owen [10][11][12] and is further generalized by Qin and Lawless [13]. Empirical likelihood method is used for statistical inference of various models (see Chen and Keilegom [14] and Nordman and Lahiri [15]). In addition, some statisticians have also begun to pay attention to the statistical inference of semiparametric regression model with constraints (see Amini and Roozbeh [16], Roozbeh and Hamzah [17], and Roozbeh et al. [18]). is paper proceeds as follows. We first introduce the methodology and the main results. Next, we will undertake a simulation study to illustrate the feasibility of our method. Lastly, we give the proofs of the main results. e symbols "⟶ d " and "⟶ p " denote convergence in distribution and convergence in probability, respectively. Convergence "almost surely" is written as "a.s." Furthermore, A ⊗ B denotes the Kronecker product of matrices A and B, ‖ · ‖ denotes Euclidean norm of the matrix or vector, A τ denotes the transposition of the matrix or the column vector A, o p (1) denotes a random variable that converges to zero in probability, and O p (1) denotes a random variable that is bounded in probability.

Methods and Main Results
In this section, we will introduce our method and give the main results of this paper. Firstly, we obtain the weights of the weighed least squares estimator based on the moment constraint E(g(X i , θ 0 )) � 0 by using the empirical likelihood method (see Owen [12]). Specifically, let where θ is unknown. Combining with the least squares method, we can obtain the following weighed least squares estimator: Introducing Lagrange multipliers λ θ 0 ∈ R r in L(θ 0 ), we obtain the following weights: where λ θ 0 satisfies erefore, by (3) and (4), we know that In order to obtain the limiting distribution of α, we assume that the following conditions hold: ere exists a neighborhood of θ 0 and an integrable function Condition (A 2 ) can be found in a study by Qin and Lawless [13]. e following theorem presents the asymptotic properties of α.

Theorem 1. Under (A 1 ) and (A 2 ) and assuming that
where ). We know that the asymptotic variance of the least squares estimator Hence, eorem 1 implies that the weighted least squares estimator has a smaller asymptotic variance compared with the least squares estimator. at is to say, using the moment restrictions does improve the effectiveness of the estimation.
α contains unknown parameters. erefore, we need to further estimate the unknown parameter θ. Let θ � arg max θ L(θ) in (2). Using results in a study by Qin and Lawless [13], we know that where λ and (λ θ , θ) solves Let ere are no unknown parameters in α ′ , so it can be used to estimate unknown parameters α in practice. Next, we investigate the limiting properties of α ′ . For convenience sake, where I is the unit matrix. e following theorem gives the limiting property of α ′ .
Similarly, since matrix B is nonnegative definite, the weighted least squares estimator α ′ also has a smaller asymptotic variance compared with the least squares estimator.

Simulation Studies
In this section, we study the finite sample properties of the above estimator by simulation. We consider the following models: where the distribution of X i is exponential with unit mean and the distribution of ε i is standard normal. 9), and the probability distribution of λ i is where the distribution of X 1i is exponential with mean 0.5, the distribution of X 2i is exponential with unit mean, and the distribution of ε i is standard normal.
For each experiment, we conduct 1000 repetitions, and three alternative sample sizes (n � 30,100 and 300) are considered.
e simulation results for model 1 are summarized in Table 1. For model 2, we take different pollution level p � 0.2, 0.4, and the simulation results are presented in Tables 2 and 3, respectively. Moreover, the simulation results for model 3 are summarized in Table 4.
It can be seen from Tables 1-4 that the ratio of the mean absolute deviations of the least squares estimation with moment restrictions to that of the ordinary least squares estimation is very small. is implies that, for the different sample sizes, the different error, and the different parameters, the least squares estimation with moment restrictions is a more precise estimator. Hence, utilizing the auxiliary information indeed improves the efficiency of estimation.

Proofs of the Main Results
In order to prove eorem 1, we first present several lemmas.
Using the method of the proof of Lemma 2 in [19], we can prove that (13) holds. Lemma 1 is established. □ Lemma 2. Under (A 1 ) and (A 2 ) and assuming that EX 4 1 < ∞, if α 0 is the true value of α, then where Proof. According to the central limit theorem of independent and identical distribution, it is easy to know that Lemma 2 holds.      Mathematical Problems in Engineering

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erefore, we have Note that Furthermore, by Lemma 1, we obtain By the law of large numbers for independent and identically distributed random variables, we have Combining with (21) and (22), we know that Lemma 3 holds.
By (5), we know that where By Lemmas 1-3, we know that By using Taylor's formula for (1/1 + λ τ θ 0 g(X i , θ 0 )), we have So, we have By Lemma 1 and Lemma 3, we have erefore, we have that uniformly for i, Note that where Firstly, we prove that After simple algebra calculation, we have By the law of large numbers for independent and identically distributed random variables, we have Furthermore, by (26) and Lemma 2, we know that Combined with (36) and (37), this establishes (33). In the following, we consider S n . By (26) and (26), we know that By the law of large numbers for independent and identically distributed random variables, we know that Furthermore, by Lemma 2, we know that Combined with (31), we know that eorem 1 holds.

□
Proof of eorem 2 Using the results in [13], we know that After simple algebra calculation, we have where Similar to the proof of eorem 1, we know that us, eorem 2 is established.

Conclusions
In this paper, we discuss how to use auxiliary information to improve the efficiency of regression model parameter estimation when auxiliary information exists. First of all, we use the auxiliary information to establish the estimation equation. Based on this estimation equation, we use empirical likelihood method to obtain the weight of weighted least squares estimation of regression model parameters and then give the weighted least squares estimation of model parameters. Secondly, it is noticed that the auxiliary information contains unknown parameters, so the weighted least squares estimator also contains unknown parameters. erefore, we use the maximum empirical likelihood estimation to further estimate the unknown parameters contained in the auxiliary information. erefore, we then obtain the weighted least squares estimation that can be used in practice. Finally, we use simulation analysis to illustrate the feasibility of our method.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this article. 6 Mathematical Problems in Engineering