Abstract
In this paper, we investigate the variable selection for varying coefficient errors-in-variables (EV) models with longitudinal data when some covariates are measured with additive errors. A variable selection method based on bias-corrected penalized quadratic inference function (pQIF) is proposed by combining the basis function approximation to coefficient functions and bias-corrected quadratic inference function (QIF) with shrinkage estimations. The proposed method can handle the measurement errors of covariates and within-subject correlation, estimate and select non-zero nonparametric coefficient functions. With appropriate selection of the tuning parameters, we establish the consistency of the variable selection method and the sparsity properties of the regularized estimators. The finite sample performance of the proposed method is assessed by simulation studies. The utility of the method is further demonstrated via a real data analysis.
Acknowledgments
We thank the editor and reviewers for their helpful comments that significantly improved the manuscript.
Appendix 1. Derivation process of EquationEquation (9)(9) (9)
According to the above, we can see that
For simplicity, denote According to for so we ca get where and
The derivation process of EquationEquation (9)(9) (9) is finished.
Appendix 2. Proof of theorems
Lemma 1.
Suppose Conditions C2 and C8 hold and , there exists a constant that satisfies (22) (22)
Lemma 1 is the Corollary 6.21 in Schumaker (Citation2007), the proof is omitted here.
Lemma 2.
Assume Conditions C1-C11 hold, and , then we have
where “” represents the convergence in probability.
where “” represents the convergence in distribution.
Proof.
We first prove part (i). According to EquationEquation (12)(12) (12) , we can get the first derivative of about β as (23) (23)
Consider the kth block matrix of as
Now, we prove as where Clearly we get as
According to EquationEquation (10)(10) (10) , we see that is the sample covariance matrix of Σu, which implies that is the mean of some sample covariance matrices and as According to the plug-in principle, we get
Under Condition C9, we can get To prove denote where we can get and
From Conditions C4-C7, is bounded. By the law of large numbers, So we get and where The proof of the part (i) is completed.
Nextly, we prove part (ii). We first prove Consider the kth block matrix of as where
Obviously, we have and
From Conditions C4 to C7, is bounded, and we can get by the law of large numbers. Similarly, we have and satisfies.
In addition, According to the Cauchy-Schwarz inequality we have
Therefore,
Under Condition C8 and Lemma 1, we have From the definition of and by the law of large numbers, By the definition of and the central limit theorem, we can get So we have and
Next, we prove According to the above conclusions, we have where Furthermore, we get
Obviously, according to the above conclusions, we can get
Under conditions C4-C7, following Tian, Xue, and Liu (Citation2014), for any which satisfies and Similarly, for any such that then Using the Cauchy-Schwarz inequality, for any such that So, we know that and satisfy the Lyapunov condition for central limit theorem. In addition, we have under condition C5, so we get such that which implies that satisfies the Lyapunov condition for the central limit theorem. Thus
According to condition C4, we have So
The proof of Lemma 2 is completed.
Lemma 3.
Suppose that the preceding regularity conditions of C1-C11 hold, then
Proof.
Following Tian, Xue, and Liu (Citation2014), apply Taylor expansion to at we have where is a three-dimensional array of By Lemma 2, we can see that Under condition C4, we have
So, we have
Similarly, we get where is a four-dimensional array
By the definition of and so we have Using Lemma 2, we get
Hence we have So we get
The proof of Lemma 3 is completed.
Proof of Theorem 1.
From Lemma 1, we have Suppose and To prove Theorem 1, we just have to show that for any there exists a large constant C such that (24) (24)
When (24) is always true. So we just assume Without loss of generality, assume and we have
Apply Taylor expansion to at we have where lies between β and According to Lemmas 1 and 2, we can get and
Therefore we have
Obviously, When C is large enough,
So when C is large enough,
Assume and When n is large enough, we have Following the definition of the penalty function, we get
So, for any given there exists a large enough C which satisfies EquationEquation (24)(24) (24) , which further implies that there exists which satisfies Note that
With the same arguments above, we can get Therefore, invoking we have With Lemma 1, we get Thus, we complete the proof of Theorem 1.
Proof of Theorem 2.
Assume for and are non-zero coefficient functions. So we get the corresponding regression parameter space as
For denote where 0 is an vector of zeros. From Lemma 1 and Xue, Qu, and Zhou (Citation2010), we have and To prove Theorem 2, it is sufficient to show that, for any and is true with probability 1. where lies between and β, t lies between and Furthermore, we get
According to Lemmas 2 and 3, we have
Form Conditions C10 and C11,
So for any and is true with probability tending to 1. This completes the proof of Theorem 2.
Proof of Theorem 3.
Following Wang, Li, and Tsai (Citation2007) and Tian, Xue, and Liu (Citation2014), we create three mutually exclusive sets: where and represent underfitted, correctly fitted or overfitted model respectively. Then, the theorem can be proved by comparing and Here we consider two separate cases.
Case I: When we have
By the law of large numbers and the continuous mapping theorem, we can get Case II: When we have and So we can get
Both cases hold true in probability by the law of large numbers and the continuous mapping theorem. This completes the proof of the Theorem 3.