Adaptive structure inferences on partially linear error-in-function models with error-prone covariates

Abstract

Model structural inference on semiparametric measurement error models have not been well developed in the existing literature, partially due to the difficulties in dealing with unobservable covariates. In this study, a framework for adaptive structure selection is developed in partially linear error-in-function models with error-prone covariates. Firstly, based on the profile-least-square estimators of the current models, we define two test statistics via generalized likelihood ratio (GLR) test method (Fan et al. in Ann Stat 29(1):153–193, 2001). The proposed test statistics are shown to possess the Wilks-type properties, and a class of new Wilks phenomenon is unveiled in the family of semiparametric measurement error models. Then, we demonstrate that the GLR statistics asymptotically follow chi-squared distributions under null hypotheses. Further, we propose efficient algorithms to implement our methodology and assess the finite sample performance by simulated examples. A real example is given to illustrate the performance of the present methodology.

Appendix

Before demonstrating proofs of the main theorems introduced in Sect. 2, we now give the following conditions used in the paper.

Condition(A)

$(A_{1})$ :

Nonparametric function $g(\cdot )$ has the Lipschitz continuous second derivative $g''(\cdot )$.

$(A_{2})$ :

The covariates $\xi$ and T have a joint density $p(\xi ,t)$. Further, the marginal density $f_{T}(t)$ of T is compactly supported, bounded, Lipschitz continuous and bounded away from 0. T has a bounded support $\mathscr {T}$ and $\xi$ has a bounded support.

$(A_{3})$ :

The kernel functions $K(\cdot )$ and $L(\cdot )$ are symmetric density functions with compact support $[-1,1]$, and they are bounded, nonnegative and Lipschitz continuous.

$(A_{4})$ :

$E(\xi |T)$, $E(\xi \xi ^{T}|T)$ and $E((\xi \xi ^{T})*(\xi \xi ^{T})|T)$ are Lipschitz continuous, where $G*H$ denotes the Hadamard product of matrices G and H.

$(A_{5})$ :

$E(\varepsilon ^{4}_{i})<\infty$.

$(A_{6})$ :

The marginal density $f_{T}(t)$ of T has the bounded kth derivative, where k is a positive integer. The random variable X has a bounded support $\mathscr {X}$. The density function $f_{X}(x)$ of X is bounded away from 0 on $\mathscr {X}$.

$(A_{7})$ :

For each $T \in \mathscr {T}$, the matrix $E(\xi \xi ^{T}|T)$ is nonsingular and the matrix $B=~E(V_{1}V_{1}^{T})$ is positive definite.

$(A_{8})$ :

Let $h_{j}(t)=E(\hat{\xi }_{ij}|T_i=t),\hat{\xi }_{ij}=\hat{\xi }_j(X_i),1\le i\le n,1\le j\le p$. Both $g(\cdot )$ and $h_{j}(\cdot )$ are Lipschitz continuous of order 1.

$(A_{9})$ :

There is a $d>2$ such that $E|\xi |^{2d}<\infty$, $E|X|^{2d}<\infty$, and for $\delta <2-d^{-1}$, $n^{2\delta -1}h\rightarrow \infty$, $n^{2\delta -1}b_{k}\rightarrow \infty$, $nhb_{k}^{2(r+1)}\rightarrow 0$, where $\{b_{k}\}^p_{k=1}$ is the bandwidth sequence.

$(A_{10})$ :

$nh^{8}\rightarrow 0$, $nh^{2}/(\log n)^{2}\rightarrow \infty$.

$(A_{11})$ :

The $\phi _{v}(\cdot )$ served as the characteristic function of v is not identically zero. And it is assumed to be ordinary smooth or super smooth.

Lemma

Suppose Conditions $(A_1)$–$(A_4)$ hold, then, uniformly in $t\in \mathscr {T}$, we have

$$\begin{aligned} \hat{g}_{n}(t)-g(t)=\frac{1}{2}\nu _{2}g''(t)h^2+\frac{1}{n} \sum ^n_{k=1}\frac{1}{f_T(t)}K_{h}(T_k-t) \hat{\varepsilon }_k+o_p\left( \frac{1}{\sqrt{nh}}+h^2\right) . \end{aligned}$$

under $H_{0}$, $t\in \mathscr {T}$, $h\rightarrow 0,~nh^{3/2}\rightarrow \infty$, thus

$$\begin{aligned} \hat{g}_{n}(t)-g(t)=\frac{1}{n}\sum ^n_{k=1} \frac{1}{f_T(t)}K_{h}(T_k-t)\hat{\varepsilon }_k +o_p\left( \frac{1}{\sqrt{nh}}\right) ~\hat{=}e(t)+o_p\left( \frac{1}{\sqrt{nh}}\right) . \end{aligned}$$

(A.1)

where $\nu _{2}=\int s^2K(s)ds$, $e(t)=\frac{1}{n}\sum ^n_{k=1}\frac{1}{f_T(t)}K_{h}(T_k-t) \hat{\varepsilon }_k$.

The lemma obtained from Carroll et al. (1997) shows the asymptotic property of $g(\cdot )$ when $\beta$ is estimated at the parametric rate. The asymptotic behaviour $\hat{\beta }_n-\beta =O_p(n^{-1/2})$ can be inferred naturally from Huang and Ding (2017).

Proof of Theorem 1

The proof of Theorem 1 is mainly based on the conclusion of Zhang (2007). Represent the GLR statistic for testing problem (3) as

$$\begin{aligned} \lambda _{ng}=\frac{\frac{1}{2}(RSS_{0g}-RSS_1)}{RSS_1/n}, \end{aligned}$$

the numerator part of $\lambda _{ng}$ can be inferred as

$$\begin{aligned} \frac{1}{2}(RSS_{0g}-RSS_1)=-\frac{1}{2}\sum ^{n}_{i=1}e^2(W_i)+\sum ^{n}_{i=1}e(W_i)\hat{\varepsilon }_i+o_p(h^{-1/2}), \end{aligned}$$

(A.2)

We prove the numerator formula (A.2) in the first place. By the following simple computations we can derive the correlative residuals sum of squares,

$$\begin{aligned} RSS_{0g}&=\sum ^{n}_{i=1}(Y_i-\check{\beta }^T\hat{\xi }_{i}-\check{\gamma }_0-\check{\gamma }_1W_i)^2\\&~=\sum ^{n}_{i=1}[(\beta ^T\hat{\xi }_i+\gamma _0+\gamma _1W_i+\hat{\varepsilon }_i)-(\check{\beta }^T\hat{\xi }_{i}+\check{\gamma }_0+\check{\gamma }_1W_i)]^2\\&~=\sum ^{n}_{i=1}[(\gamma _0-\check{\gamma }_0)+(\gamma _1-\check{\gamma }_1)W_i+(\beta -\check{\beta })^T\hat{\xi }_i+\hat{\varepsilon }_i]^2\\&~=\sum ^{n}_{i=1}\hat{\varepsilon }^2_i+O_p(1), \end{aligned}$$

Analogously, we can get

$$\begin{aligned} RSS_1&=\sum ^{n}_{i=1}(Y_i-\hat{\beta }^T_n\hat{\xi }_{i}-\hat{g}_n(W_i))^2\nonumber \\&~=\sum ^{n}_{i=1}[(\beta ^T\hat{\xi }_i+g(W_i)+\hat{\varepsilon }_i)-(\hat{\beta }^T_n\hat{\xi }_{i}+\hat{g}_n(W_i))]^2\\&~=\sum ^{n}_{i=1}[(g(W_i)-\hat{g}_n(W_i))+(\beta -\hat{\beta }_n)^T\hat{\xi }_i+\hat{\varepsilon }_i]^2. \end{aligned}$$

Let $A=g(W_i)-\hat{g}_n(W_i)$, $B=(\beta -\hat{\beta }_n)^T\hat{\xi }_i$, $C=\hat{\varepsilon }_i$, by (A.1), we have

$$\begin{aligned} \sum A^2= & {} \sum ^{n}_{i=1}[g(W_i)-\hat{g}_n(W_i)]^2=\sum ^{n}_{i=1}e^2(W_i)+o_p(h^{-1/2}),\\ \sum AC= & {} \sum ^{n}_{i=1}[g(W_i)-\hat{g}_n(W_i)]\hat{\varepsilon }_i=-\sum ^{n}_{i=1}e(W_i)\hat{\varepsilon }_i+o_p(h^{-1/2}). \end{aligned}$$

The previously mentioned formula $\hat{\beta }_n-\beta =O_p(n^{-1/2})$ concerned with asymptotic property of $\beta$ results in

$$\begin{aligned} \sum B^2=\sum ^{n}_{i=1}[(\beta -\hat{\beta }_n)^T\hat{\xi }_i]^2=O_p(1),~\sum BC=\sum ^{n}_{i=1}[(\beta -\hat{\beta }_n)^T\hat{\xi }_i]\hat{\varepsilon }_i=O_p(1). \end{aligned}$$

Applying the Kolmogorov’s strong law of large numbers, we can obtain

$$\begin{aligned} \sum AB=\sum ^{n}_{i=1} [g(W_i)-\hat{g}_n(W_i)] [(\beta -\hat{\beta }_n)^T\hat{\xi }_i]=O_p(1). \end{aligned}$$

Then,

$$\begin{aligned} RSS_1&~=\sum (A^2+B^2+C^2)+2\sum (AB+AC+BC)\\&~=\sum ^{n}_{i=1}e^2(W_i)-2\sum ^{n}_{i=1}e(W_i) \hat{\varepsilon }_i+\sum ^{n}_{i=1}\hat{\varepsilon }^2_i+o_p(h^{-1/2}). \end{aligned}$$

The combination of the expressions of $RSS_{0g}$ and $RSS_1$ accounts for the establishment of the formula (A.2). According to Zhang (2007), we further deduce

$$\begin{aligned} \frac{1}{2}(RSS_{0g}-RSS_1)&=\frac{1}{n} {\mathop {\sum }^{n}_{\begin{array}{c} i,k=1\\ i\ne k \end{array}}}\frac{\hat{\varepsilon }_k\hat{\varepsilon }_i}{f_T(W_k)}\left( K_h(W_k-W_i)-\frac{1}{2}K_h*K_h(W_k-W_i)\right) \nonumber \\&\quad +\frac{\sigma ^2|\mathscr {T}|}{h}\left( K(0)-\frac{1}{2}\int K^2(s)ds\right) +o_p\left( \frac{1}{\sqrt{h}}\right) \nonumber \\&=W_n+\sigma ^2\mu _n+o_p\left( \frac{1}{\sqrt{h}}\right) , \end{aligned}$$

(A.3)

where,

$$\begin{aligned} W_n&=\frac{1}{n}{\mathop {\sum }^{n}_{\begin{array}{c} i,k=1\\ i\ne k \end{array}}}\frac{\hat{\varepsilon }_k\hat{\varepsilon }_i}{f_T(W_k)}\left( K_h(W_k-W_i)-\frac{1}{2}K_h*K_h(W_k-W_i)\right) ,\\ \mu _n&=\frac{|\mathscr {T}|}{h}\left( K(0)-\frac{1}{2}\int K^2(s)ds\right) . \end{aligned}$$

To demonstrate the formula (A.3), we combine the arguments of Zhang (2007) and (A.1), then

$$\begin{aligned} \frac{1}{2}(RSS_{0g}-RSS_1)&= S_1-S_2, \end{aligned}$$

and,

$$\begin{aligned} S_1&=\sigma ^2\frac{K(0)|\mathscr {T}|}{h}+\frac{1}{n}{\mathop {\sum }^{n}_{\begin{array}{c} i,k=1\\ i\ne k \end{array}}} \frac{\hat{\varepsilon }_i\hat{\varepsilon }_k}{f_T(W_i)}K_h(W_k-W_i) +O_p\left( \frac{1}{\sqrt{nh^2}}\right) ,\\ S_2&\equiv S_{21}+S_{22}+S_{23}+S_{24}\\&=\frac{1}{2n}{\mathop {\sum }^{n}_{\begin{array}{c} j,k=1\\ j\ne k \end{array}}} \frac{\hat{\varepsilon }_k\hat{\varepsilon }_j}{f_T(W_k)h}\int K(s)K\left( s-\frac{W_j-W_k}{h}\right) ds\\&\quad +\frac{\sigma ^2|\mathscr {T}|}{2h}\int K^2(s)ds+O_p\left( 1+\frac{1}{\sqrt{nh^2}} +\frac{1}{nh^2}+\frac{1}{\sqrt{n^2h^3}}\right) , \end{aligned}$$

where,

$$\begin{aligned} S_{21}&=\frac{1}{2n^2}\sum ^n_{i=k}\frac{\hat{\varepsilon }^2_i}{f^2_T(W_i)}K^2_h(0)=O_p\left( \frac{1}{nh^2}\right) ,\\ S_{22}&=\frac{\sigma ^2|\mathscr {T}|}{2h}\int K^2(s)ds+O_p\left( 1+\frac{1}{\sqrt{nh^2}}+\frac{1}{\sqrt{n^2h^3}}\right) ,\\ S_{23}&=O_p\left( \frac{1}{\sqrt{n^2h^3}}\right) ,\\ S_{24}&=\frac{1}{2n}{\mathop {\sum }^{n}_{\begin{array}{c} j,k=1\\ j\ne k \end{array}}}\frac{\hat{\varepsilon }_k\hat{\varepsilon }_j}{f_T(W_k)h}\int K(s)K\left( s-\frac{W_j-W_k}{h}\right) ds +O_p\left( 1+\frac{1}{\sqrt{nh^2}}\right) . \end{aligned}$$

Therefore, the numerator part (A.3) of the GLR statistic $\lambda _{ng}$ is vindicated. By Zhang (2007), the variance of $W_n$ can be written as

$$\begin{aligned} Var(W_n)=\sigma ^4\sigma ^2_n(1+o(1)), \end{aligned}$$

where $\sigma ^2_n=\frac{2|\mathscr {T}|}{h}\int (K(s)-\frac{1}{2}K*K(s))^2ds$. It implies that

$$\begin{aligned} W_n=O_p(\sqrt{1/h}). \end{aligned}$$

(A.4)

Conjoining with (A.3) with (A.4), we obtain the denominator part of $\lambda _{ng}$,

$$\begin{aligned} \frac{RSS_1}{n}=\frac{RSS_{0g}}{n}-\frac{2\sigma ^2\mu _n}{n}-\frac{2W_n}{n}+o_p\left( \frac{1}{n\sqrt{h}}\right) =\sigma ^2(1+o_p(1)). \end{aligned}$$

(A.5)

We combine the numerator part (A.3) and the denominator part (A.5) of the statistic to derive

$$\begin{aligned} \lambda _{ng}=\frac{\frac{1}{2}(RSS_{0g}-RSS_1)}{RSS_1/n}=\mu _n+\frac{W_n}{\sigma ^2}+o_p\left( \frac{1}{\sqrt{h}}\right) . \end{aligned}$$

(A.6)

De Jong (1987) demonstrated that $W_n$ is asymptotically normal,

$$\begin{aligned} \frac{W_n}{\sigma ^2\sigma _n} \xrightarrow {L}N(0,1), \end{aligned}$$

then we can get

$$\begin{aligned}\frac{1}{\sigma _n}(\lambda _{ng}-\mu _n)\xrightarrow {L}N(0,1), \quad i.e. \quad r_K\lambda _{ng}\sim _a\chi ^2(r_Kc_K|\mathscr {T}|/h).\end{aligned}$$

This completes the proof of Theorem 1. $\square$

Proof of Theorem 2 and 3

Primarily, we have (A.5): $n^{-1}RSS_{1}=\sigma ^{2}(1+o_{P}(1))$, and $\hat{g}_{i0}=\sum \nolimits _{j=1}^{n}\omega _{nj}(W_{i})(Y_{j}-\hat{\beta }_{0}^{T}\hat{\xi }_{j})$, $\hat{g}_{in}=\sum \nolimits _{j=1}^{n}\omega _{nj}(W_{i})(Y_{j}-\hat{\beta }^{T}_n\hat{\xi }_{j})$, then

$$\begin{aligned} RSS_{0\beta }&=\sum \limits _{i=1}^{n}(Y_{i}-\hat{g}_{i0}-\hat{\beta }_{0}^{T}\hat{\xi }_{i})^2 =\sum \limits _{i=1}^{n}\left( Y_{i}-\hat{g}_{in}-\hat{\beta }^{T}_n\hat{\xi }_{i}+(\hat{\beta }_n-\hat{\beta }_{0})^{T}\tilde{\xi }_{i}\right) ^{2} \nonumber \\&=RSS_{1}+Q_{1}+Q_{2}+Q_{3}, \end{aligned}$$

where

$$\begin{aligned} Q_{1}&=\{\tilde{\xi }(\hat{\beta }_n-\hat{\beta }_{0})\}^{T}\{\tilde{\xi }(\hat{\beta }_n-\hat{\beta }_{0})\}, \\ Q_{2}&=(Y-\hat{g}_{n}-\hat{\beta }_n^{T}\hat{\xi })\{\tilde{\xi }(\hat{\beta }_n-\hat{\beta }_{0})\},\\ Q_{3}&=\{\tilde{\xi }(\hat{\beta }_n-\hat{\beta }_{0})\}^{T}(Y-\hat{g}_{n}-\hat{\beta }_n^{T}\hat{\xi }). \end{aligned}$$

Then we can obtain

$$\begin{aligned} Q_{1}=\{\tilde{\xi }(\hat{\beta }_n-\hat{\beta }_{0})\}^{T}\{\tilde{\xi }(\hat{\beta }_n-\hat{\beta }_{0})\} =\hat{\beta }_n^{T}A_0^{T}\{A_0(\tilde{\xi }^{T}\tilde{\xi })^{-1}A_0^{T}\}^{-1}A_0\hat{\beta }_n. \end{aligned}$$

According to the Lemma 5 in Huang and Ding (2017),

$$\begin{aligned} n^{-1}\tilde{\xi }^{T}\tilde{\xi }\rightarrow B, \end{aligned}$$

thus,

$$\begin{aligned} Q_{1}-n\beta ^{T}A_0^{T}(A_0B^{-1}A_0^{T})^{-1}A_0\beta \rightarrow \sigma ^{2}\times \sum \limits _{i=1}^{l}\omega _{i}\chi _{i1}^{2}, \end{aligned}$$

(A.7)

$Q_{2},Q_{3}$ are negligible in probability. Combing (A.7) and Slutsky’s theorem, we can get

$$\begin{aligned} 2\lambda _{n\beta }-n\sigma ^{-2}\beta ^{T}A_0^{T}(A_0B^{-1}A_0^{T})^{-1}A_0\beta \xrightarrow {L}\sum \limits _{i=1}^{l}\omega _{i}\chi _{i1}^{2}. \end{aligned}$$

We can learn from Rao and Scott (1981) that the distribution $\varrho _{n}\sum \nolimits _{i=1}^{l}\omega _{i}\chi _{i1}^{2}$ is nearly the same as the $\chi ^2$ distribution with degrees of freedom l. The proof of Theorem 3 is completed. $\square$

The Theorem 2 is the special case of the Theorem 3. Therefore, the proof of Theorem 2 can be completed by the similar arguments and the details are omitted.