Shrinkage Estimation of Semiparametric Model with Missing Responses for Cluster Data

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.


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 ( ) where ij Y , ij Z and ij X stand for the ith observation of Y, Z and X in the jth cluster.
( ) 1 , , is a vector of q-dimensional unknown parametrics; ε is random error with mean zero and variance 2 σ .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 δ .When 0 δ = , Y represents the situation of missing, and 1 δ = , otherwise.Suppose that responses are missing at random, δ 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.

Model with Complete-Case Data
Due to there exist missing responses, for simplicity, focus on the case where 1 δ = .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 , 1, , j n j m =  .For the ith subjects from the jth cluster, let

{ }
, , , , , be a set of random sample from model ( 1), then it is easy to obtain: ( ) In this situation, if the parametric component j β is given, model ( 2) can be written as: where is unknown but smooth function in u and its true value is denoted by . Suppose that the first in- predictors are relevant and the rest are not.

The Kernel Least Absolute Shrinkage and Selection Operator Method
Similarity, consider the jth cluster data firstly, given any index value , the estimator of ( ) , can be obtained by minimizing the following locally weighted least squares function: According to ( ) . Furthermore, B  is also the minimizer of the following global least squares function: with respect to , , , , thus, depended on the normal equation ( ) ≤ ≤ , its minimizer is obtained.
From another aspect, for Q(B), as one can see, tj ; see (4).In that case, B  is also the minimizer of (5).
Due to it is assumed that the last ( ) 0 p p − columns of 0 B matrix should be 0. Therefore, the goal of variable selection amounts to identifying sparse columns in matrix 0 B .In order to discriminate irrelevant variable, which implies that one should identify matrix sparse solutions in 0 B 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. ( ˆˆ, , , , , , is the kth column of B, and .means the usual Euclidean norm.

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 B  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 where ( ) m D is a p p × diagonal matrix with its kth diagonal component given by ˆjk Furthermore, for each cluster and each group, by using weighted mean idea to gain the finally estimator of coefficient vector ( ) U α .That is, the finally estimator of ( ) , , , where ( ) α in sth cluster of ith subject.

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 β are also derived.As is shown: where j β is a coefficient vector of size q.Under its assumption, there are 0 q predictors relevant and the rest are not.Similarity, following the idea of local quadratic approximation and weighted mean the finally estimator of β is given by ( ) , , .

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 Ω .Its density function

( )
. f is Lipschitz continuous and bounded away from 0 on its support.
With the purpose of considering the oracle property, define the orale estimators as follows: Suppose that the assumptions are satisfied, if n b → ∞ , 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: where jk α  is the kth column of the unpenalized estimate B  in jth cluster.Since jk α is an estimator with 0 jk λ = , the results of Theorem 1 and Theorem 2 can be applied.Thus, as long as ( ) where j df λ is the number of varying coefficients identified by Obviously, the effective sample size j n h is used instead of the original sample size j n .Further then, the tuning parameter can be given by ˆarg min .

Assumption 2 . 1 .
For each U ∈ Ω , ( )T | E ZZ U is non-singular.( ) T | E XX U , ( ) T |E ZZ U and ( )T | E XZ Uare all Lipschitz continuous.And they have bounded second order derivatives on [0derivatives in U ∈ Ω .Assumption 5.The function K(.) is a symmetric density function with compact support.Lemma Suppose that the Assumptions of (A1)-(A5) hold, The proof of Lemma 1 and Lemma 2 can be shown in Wang and Xia[25].
∞ , then we problem regarding 0 R λ ∈ .According to BIC-type criterion, 0 λ is defined as follows:

1 .
Due to each diagonal component of m aa D must converge to some finite number while m bb D diverge to infinity in the case where m → ∞ , thus each component of ( ) ( ) on [0, 1] as m → ∞ .It is easy to see that eigenvalue of an arbitrary positive definite matrix A. Notice that it completes the proof of Theorem 2. 