Received 23 September 2015; accepted 21 December 2015; published 24 December 2015

ABSTRACT

This paper simultaneously investigates variable selection and imputation estimation of semiparametric partially linear varying-coefficient model in that case where there exist missing responses for cluster data. As is well known, commonly used approach to deal with missing data is complete-case data. Combined the idea of complete-case data with a discussion of shrinkage estimation is made on different cluster. In order to avoid the biased results as well as improve the estimation efficiency, this article introduces Group Least Absolute Shrinkage and Selection Operator (Group Lasso) to semiparametric model. That is to say, the method combines the approach of local polynomial smoothing and the Least Absolute Shrinkage and Selection Operator. In that case, it can conduct nonparametric estimation and variable selection in a computationally efficient manner. According to the same criterion, the parametric estimators are also obtained. Additionally, for each cluster, the nonparametric and parametric estimators are derived, and then compute the weighted average per cluster as finally estimators. Moreover, the large sample properties of estimators are also derived respectively.

Keywords:

Semiparametric Partially Linear Varying-Coefficient Model, Missing Responses, Cluster Data, Group Lasso

1. Introduction

In real application, the analysis of cluster data arises in various research areas such as biomedicine and so on. Without loss of generality, the data are clustered into classes in terms of the objects which have certain similar property. For example, focus on the same confidence interval as a cluster. Numerous parametric approaches are applied to the analysis of cluster data, and with the rapid development of computing techniques, nonparametric and semiparametric approaches have attained more and more interest. See the work of Sun et al. [1] , Cai [2] , Vichi [3] , Yi et al. [4] , Carrol [5] , and He [6] , among others.

Consider the semiparametric partially linear varying-coefficient model which is a useful extension of partially linear regression model and varying-coefficient model over all clusters, it satisfies

(1)

where, and stand for the ith observation of Y, Z and X in the jth cluster. is a vector of q-dimensional unknown parametrics; is a p-dimensional unknown coefficient vector. is random error with mean zero and variance.

Obviously, when m = 1, model (1) reduces to semiparametric partially linear varying-coefficient model. A series of literature (You and Chen [7] , Fan and Huang [8] , Wei and Wu [9] , Zhang and Lee [10] ) have provided the corresponding statistic inference of such semiparametric model. In [8] , Fan and Huang put forward a profile least square technique and propose generalized likelihood ratio test. In [7] , You and Chen study the estimation problem when some covariates are measured with additive errors. When m = 1 and Z = 0, model (1) becomes varying-coefficient model which has been widely studied by many authors such as Fan and Zhang [11] , Hastile and Tibshirani [12] , Xia and Li [13] , Hoover et al. [14] . When m = 1, p = 1 and Z = 1, model (1) reduces to partially linear regression model which is proposed by Engle et al. [15] when they research the influence of weather on electricity demand. See the literature of Yatchew [16] , Spechman [17] and Liang et al. [18] , among others.

However, in practice, responses may often not be available completely because of various factors. For example, some sampled units are unwilling to provide the desired information, and some investigators gather incorrect information caused by careless and so on. In that case, a commonly used technique is to introduce a new variable. When, Y represents the situation of missing, and, otherwise. Suppose that responses are missing at random, and Y are conditionally independent, then it has

Due to the practicability of the missing responses estimation, semiparametric partially linear varying-coefficient model with missing responses has attracted many authors’ attention, such as Chu and Cheng [19] , Wei [20] , Wang et al. [21] and so on.

It is worth pointing out that there is little work concerning both missing and cluster data especially in semiparametric partially linear varying-coefficient model. If ignore the difference of clusters, it leads the predictors of response values Y far away from the true values and the estimators have poor robustness. Therefore, it is necessary to take cluster data into consideration with the purpose of improving estimation efficiency. For each cluster, introduce group lasso to semiparametric partially linear varying-coefficient model respectively on the basis of complete case data. In order to automatically select variables and conduct estimation simultaneously, lasso is a popular technique which has attracted many authors’ attention such as Tibshirani [22] , Zou [23] and so on. Due to the idea of lasso is to select individual derived input variable rather than the strength of groups of input variables, in this situation, it leads to select more factors as the approach of group lasso. As is shown in Yuan and Yi [24] , Wang and Xia [25] , Hu and xia [26] and so on. Thus, this paper centers on the technique of group lasso in a computationally efficient manner. Further then, parametric and nonparametric components are obtained by computing the weighted average per cluster. As for the inference of estimators, the properties of asymptotic normality and consistency are also provided. And Bayesian information criterion (BIC) as tuning parameter selection criterion is used in this article.

The rest of the paper is organized as follows. The use of the applied method is given in Section 2. In Section 3, the theoretical properties are provided. Conclusions are shown in Section 4. Finally, the proofs of the main results are relegated to Appendix.

2. Semiparametric Model with the Methodology

2.1. Model with Complete-Case Data

Due to there exist missing responses, for simplicity, focus on the case where. That is so-called the method of complete case data. It is assumed that there are m independent clusters, and the number of observations in the jth cluster is. For the ith subjects from the jth cluster, let

be a set of random sample from model (1), then it is easy to obtain:

(2)

In this situation, if the parametric component is given, model (2) can be written as:

(3)

where. The coefficient vector is unknown but smooth function in u and its true value is denoted by. Suppose that the first integer predictors are relevant and the rest are not.

2.2. The Kernel Least Absolute Shrinkage and Selection Operator Method

Similarity, consider the jth cluster data firstly, given any index value, the estimator of, namely, can be obtained by minimizing the following locally weighted least squares function:

(4)

According to, define and. It is clear that, is a nature estimator for. Furthermore, is also the minimizer of the following global least squares function:

(5)

with respect to. Due to Q(B) is a quadratic function in B, thus, depended on the normal equation for every, its minimizer is obtained. From another aspect, for Q(B), as one can see, is only involved in; see (4). Then it satisfies, leading to the solution; see (4). In that case, is also the minimizer of (5).

Due to it is assumed that the last columns of matrix should be 0. Therefore, the goal of variable selection amounts to identifying sparse columns in matrix. In order to discriminate irrelevant variable, which implies that one should identify matrix sparse solutions in in a column-wise manner. Based on the group lasso idea of Yuan and Lin [24] , Wang and Xia [26] , the penalized estimate is shown as follows:

where is the tuning parameter.

(6)

(7)

is the kth column of B, and means the usual Euclidean norm.

2.3. Local Quadratic Approximation

It is well known that, there exist many computational algorithms for the lasso-type problems such as local quadratic approximation, the least angle regression and many others. For simplicity, this article describes here an easy implementation based on the idea of the local quadratic approximation. Specifically, the implementation is based on an iterative algorithm with as the initial estimator. Let

be the KLASSO estimate obtained in the mth iteration j cluster. Then, the loss function in (6) can be locally approximated by

whose minimizer is given by with the th row given by

(8)

where is a diagonal matrix with its kth diagonal component given by,.

Furthermore, for each cluster and each group, by using weighted mean idea to gain the finally estimator of coefficient vector. That is, the finally estimator of can be given by

where means in sth cluster of ith subject.

2.4. Estimation of Parametric Component

In terms of the above estimator of nonparametric component and according to the same criterion, the lasso estimation of parametric components are also derived. As is shown:

(9)

where is a coefficient vector of size q. Under its assumption, there are predictors relevant and the rest are not. Similarity, following the idea of local quadratic approximation and weighted mean the finally estimator of is given by

(10)

3. Theoretical Properties

3.1. Technical Conditions

The following assumptions are needed to prove the theorems for the proposed estimation methods.

Assumption 1. The random variable U has a bounded support. Its density function is Lipschitz continuous and bounded away from 0 on its support.

Assumption 2. For each, is non-singular., and are all Lipschitz continuous. And they have bounded second order derivatives on [0, 1].

Assumption 3. There is an such that and and for some such that.

Assumption 4. have continuous second derivatives in.

Assumption 5. The function K(.) is a symmetric density function with compact support.

Lemma 1. Suppose that the Assumptions of (A1)-(A5) hold, , and, then it satisfies

Lemma 2. If (A1)-(A5), , , and then for any.

The proof of Lemma 1 and Lemma 2 can be shown in Wang and Xia [25] .

3.2. Basic Theorems

Suppose that the Assumptions (A1)-(A5) hold. For j th cluster, let and Denote

Theorem 1. Assume (A1)-(A5), , , and, then we have for any

With the purpose of considering the oracle property, define the orale estimators as follows:

Theorem 2. Suppose that the assumptions are satisfied, if, , and, then it is easy to see that

3.3. Tuning Parameter Selection

In the case where and, the optimal convergence rata can be obtained and the true

model can be consistently identified. Due to there exists a great challenge to select p shrinkage parameters, thus as shown in Zou [23] , wang and xia [25] , simplify the tuning parameters as follows:

(11)

where is the kth column of the unpenalized estimate in jth cluster. Since is an estimator with, the results of Theorem 1 and Theorem 2 can be applied. Thus, as long as but, one can conclude that and. Furthermore, the original p-dimensional problem about becomes a univariate problem regarding. According to BIC-type criterion, is defined as follows:

(12)

where is the number of varying coefficients identified by. is

(13)

Obviously, the effective sample size is used instead of the original sample size. Further then, the tuning parameter can be given by

Note that as an arbitrary model with a total of nonzero coefficients (i.e.). Then, means the true model and denotes the model identified by the proposed estimate. Consequently, represents the model identified by.

Theorem 3. Selection Consistency. Suppose that Assumptions (A1)-(A5) hold, the tuning parameter selected by the BIC criterion can indeed identify the true model consistency, i.e. as.

4. Conclusion

In this paper, it mainly discusses the shrinkage estimation of semiparametric partially linear varying-coefficient model under the circumstance that there exist missing responses for cluster data. Combined the idea of complete-case data, this paper introduces group lasso into semiparametric model with different cluster respectively. The new method simultaneously conducts variable selection and model estimation. Meanwhile, the technique not only reduces biased results but also improves the estimation efficiency. Finally, combined the idea of weighted mean, the nonparametric and parametric estimators are derived. The BIC criterion as tuning parameter selection is well applied in this artice. Furthermore, the properties of asymptotic normality and consistency are also derived theoretically.

Acknowledgements

This work is supported by the National Natural Science Foundation of China (61472093). This support is greatly appreciated.

Cite this paper

MingxingZhang,JiannanQiao,HuaweiYang,ZixinLiu, (2015) Shrinkage Estimation of Semiparametric Model with Missing Responses for Cluster Data. Open Journal of Statistics,05,768-776. doi: 10.4236/ojs.2015.57076

References

Appendix

Proof of Theorem 1

Proof. Based on Lemma 2 and as shown in Hunter and Li [27] , one can know that for each. Then, as long as, one can see that when then converge to a positive number, otherwise, converge to 0. Denote as the upper diagonal submatrix of and

as the lower diagonal submatrix of. From the definition of, it is remarkable that each diagonal component of must converge to some finite number while diverge to infinity in the case where.

For simplify, we follow (8) and can be rewritten as, where is a block matrix given by and with and If is given by one obtains

Due to each diagonal component of must converge to some finite number while diverge to infinity in the case where, thus each component of and converge to 0 uniformly on [0, 1] as. It is easy to see that

where, and are uniformly bounded. Obviously, as when. Therefore, sup for every. It completes the proof of Theorem 1.

Proof of Theorem 2

Proof. As is well known, is the solution of the following equation

That is to say, satisfies

where. By Lemma 2 and combined with the oracle estimator, it satisfies

where with represents the maximal eigenvalue of an arbitrary positive definite matrix A. Notice that as a result it completes the proof of Theorem 2. W