A new estimation in functional linear concurrent model with covariate dependent and noise contamination

Abstract

Functional linear concurrent regression model is an important model in functional regression. It is usually assumed that realizations of functional covariate are independent and observed precisely. But in practice, the dependence across different functional sample curves often exists. Moreover, each realization of functional covariate may be contaminated with noise. To address this issue, we propose a novel estimation method, which makes full use of dependence information and filters out the impact of measured noise. Then, we extend the proposed method to partially observed functional data. Under some regular conditions, we establish asymptotic properties of the estimators of the model. Finite-sample performance of our estimation is illustrated by Monte Carlo simulation studies and a real data example. Numerical results reveal that the proposed method exhibits superior performance compared with the existing methods.

Appendix

In this section, we give the proof of Theorems 1 and 2. The proof of theorems will require some Lemmas. We first list several lemmas that are necessary to prove the theorems.

Lemma 1

Let $B_1(t),\dots ,B_{K}(t)$ be a set of B-spline basis function defined on compact interval ${\mathcal {T}}$. For any spline function $\sum _{k=1}^{K}b_kB_K(t)$, there exist two positive constants $M_1$ and $M_2$ satisfying that

$$\begin{aligned} \frac{M_1\Vert \varvec{b}\Vert ^2}{K} \le \int _{{\mathcal {T}}} \left[ \sum _{k=1}^{K}b_kB_K(t)\right] ^2 \mathrm{{d}}t \le \frac{M_2\Vert \varvec{b}\Vert ^2}{K}, \end{aligned}$$

where $\Vert \varvec{b}\Vert ^2=\sum _{k=1}^{K}b_k^2$.

Proof

This lemma comes from Theorem 4.2 of Chapter 5 of DeVore and Lorentz (1993). $\square$

Lemma 2

Let ${\mathcal {L}}_s({\mathcal {F}})$ denote the class of ${\mathcal {F}}$-measurable random process X(t) satisfying $\Vert X\Vert _{s}=\left( \int (\textrm{E}|X(t)|^s\mathrm{{d}}t)\right) ^{1/s}<\infty$. Assume X(t) and Y(t) are two random processes such that $X(t)\in {\mathcal {L}}_s(\sigma (X(t)))$ and $Y(t)\in {\mathcal {L}}_s(\sigma (Y(t)))$. Let p, q and r be three positive numbers such that $1/p+1/q+1/r=1$. Then, there exist a constant C such that

$$\begin{aligned} \int |\textrm{Cov}(X(t),Y(t))|\mathrm{{d}}t \le C \Vert X\Vert _{p}\Vert Y\Vert _{q}\alpha \left( \sigma (X(t)),\sigma (Y(t))\right) ^{1/r}, \end{aligned}$$

where $\alpha \left( \sigma (X(t)),\sigma (Y(t))\right)$ denotes $\alpha$-mixing coefficients between two $\sigma (X(t))$ and $\sigma (Y(t))$.

Proof

This lemma comes from Lemma 2.1 of Davydov (1968). $\square$

Lemma 3

Suppose that Conditions C1–C2 hold, then we have

$$\begin{aligned} \textrm{E}\left\| \frac{1}{\sqrt{n-m}}\sum _{i=1}^{n-m}W_i^c\right\| ^2<\infty , \end{aligned}$$

Proof

This lemma comes from Theorem 2.17 of Bosq (2012). $\square$

Lemma 4

Suppose that Conditions C1–C2 hold, then for any $m=1,\dots ,M$, it holds that

$$\begin{aligned} \textrm{E}\left\| \frac{1}{\sqrt{n-m}}\sum _{i=1}^{n-m}\left[ W_i^cW_{i+m}^c-\textrm{E}\left( W_i^cW_{i+m}^c\right) \right] \right\| ^2<\infty . \end{aligned}$$

Proof

Let

$$\begin{aligned} A_k(t)=\frac{1}{\sqrt{n-m}}\sum _{i=1}^{n-m}\left[ W_i^c(t)W_{i+m}^c(t)-\textrm{E}\left( W_i^c(t)W_{i+m}^c(t)\right) \right] , m=1,\dots ,M. \end{aligned}$$

Then,

$$\begin{aligned} \textrm{E}\Vert A_k\Vert ^2= & {} \frac{\sum _{i=1}^{n-m}\textrm{E}\left\| W_i^cW_{i+m}^c\right\| ^2+2\sum _{1\le i<j \le n-m}\int \textrm{Cov}(W_i^cW_{i+m}^c,W_j^cW_{j+m}^c)}{n-m} \nonumber \\= & {} \frac{B_1+B_2}{n-m} \end{aligned}$$

(15)

Using Hölder inequality, we deduce that

$$\begin{aligned} B_1 \le \sum _{i=1}^{n-m} \textrm{E}\left\| (W_i^c)^2\right\| \textrm{E}\left\| (W_{i+m}^c)^2\right\| \le \sum _{i=1}^{n-m} \textrm{E}\Vert (W_i^c)^2\Vert ^2 \end{aligned}$$

By Condition C2, there exist a constant $C_1>0$ such that

$$\begin{aligned} B_1 \le C_1(n-m). \end{aligned}$$

(16)

Let

$$\begin{aligned} B_2= & {} 2\sum _{i=1}^{n-m}\sum _{j=i+1}^{i+m} \int \textrm{Cov}(W_i^cW_{i+m}^c,W_j^cW_{j+m}^c)\nonumber \\{} & {} + 2\sum _{i=1}^{n-m}\sum _{j=i+m+1}^{n-m} \int \textrm{Cov}(W_i^cW_{i+m}^c,W_j^cW_{j+m}^c) \nonumber \\= & {} B_{21}+B_{22}. \end{aligned}$$

(17)

Note that

$$\begin{aligned} B_{21}\le & {} 2\sum _{i=1}^{n-m}\sum _{j=i+1}^{i+m} \left( \int \textrm{E}|W_i^cW_{i+m}^cW_j^cW_{j+m}^c| +\int |\textrm{E}(W_i^cW_{i+m}^c)\textrm{E}(W_j^cW_{j+m}^c)|\right) \nonumber \\\le & {} 2\sum _{i=1}^{n-m}\sum _{j=i+1}^{i+m}\left( \textrm{E}\int (W_i^c)^4+\int \left( \textrm{E}(W_i^c)^2\right) ^2\right) \le 4C_1(n-m)m. \end{aligned}$$

(18)

By Lemma 2 and Condition C1, there exist constants $C_2,C_3>0$ such that

$$\begin{aligned} B_{22}\le & {} \sum _{i=1}^{n-m}\sum _{j=i+m+1}^{n-m} 2C \Vert W_i^cW_{i+m}^c\Vert _{4}\Vert W_j^cW_{j+m}^c\Vert _{4}\alpha _{j-i-m}^{1/2} \nonumber \\\le & {} \sum _{i=1}^{n-m}\sum _{j=i+m+1}^{n-m} 2C \int (\textrm{E}(W_i^c)^8)^{1/2}\alpha _{j-i-m}^{1/2} \nonumber \\\le & {} \sum _{i=1}^{n-m}\sum _{k=1}^{\infty } C_2 \alpha _{k}^{1/2} < C_3(n-m) \end{aligned}$$

(19)

Combining (15)–(19), there exist a constant $C_4>0$ such that

$$\begin{aligned} \textrm{E}\Vert A_k(t)\Vert ^2\le C_1+4C_1m+C_3=C_4<\infty . \end{aligned}$$

The proof of Lemma 4 is completed. $\square$

Lemma 5

Suppose that Conditions C1–C3 hold, then there exist two positive constants $c_3$ and $c_4$ such that, except on an event whose probability tends to zero, all the eigenvalues of $K\int _{T_0}^{T_1}{\hat{\varvec{G}}}(t)^T{\hat{\varvec{G}}}(t)\mathrm{{d}}t$ fall between $c_3$ and $c_4$.

Proof

Let $K\int _{T_0}^{T_1}{\hat{\varvec{G}}}(t)^T{\hat{\varvec{G}}}(t)\mathrm{{d}}t=\varvec{D}$. Let $\lambda _{\max }$ and $\lambda _{\min }$ be the maximum and minimum eigenvalues of $\varvec{D}$ respectively. Then, we deduce that

$$\begin{aligned} \lambda _{\max }=\max _{\Vert \theta \Vert =1}\theta ^T \varvec{D}\theta ,\quad \lambda _{\min }=\min _{\Vert \theta \Vert =1}\theta ^T \varvec{D}\theta . \end{aligned}$$

(20)

Let

$$\begin{aligned} \theta ^T \varvec{D}\theta =\theta ^T (\mathrm{E\varvec{D}}) \theta +\left( \theta ^T \varvec{D}\theta -\theta ^T(\mathrm{E\varvec{D}})\theta \right) . \end{aligned}$$

(21)

In the first term, according to Lemma 3, we have

$$\begin{aligned} \theta ^T (\mathrm{E\varvec{D}}) \theta= & {} K\sum _{k=1}^{K}\theta _{k}\sum _{l=1}^{K}\theta _{l}\sum _{m=1}^{M}\left( \int _{T_0}^{T_1}(\textrm{E}W_i^cW_{i+m}^c)^2 B_k(t)B_l(t)\mathrm{{d}}t\right) \\= & {} K\sum _{k=1}^{K}\theta _{k}\sum _{l=1}^{K}\theta _{l}\sum _{m=1}^{M}\left( \int _{T_0}^{T_1}(\gamma _m(t))^2 B_k(t)B_l(t)\mathrm{{d}}t\right) +o_p(1). \end{aligned}$$

By Condition C3 and Lemma 1, we deduce that

$$\begin{aligned} c_3|\theta \Vert ^2\le \theta ^T (\mathrm{E\varvec{D}}) \theta \le c_4|\theta \Vert ^2. \end{aligned}$$

(22)

For the second term, By Lemma 4, we have

$$\begin{aligned} \theta ^T \varvec{D}\theta -\theta ^T(\mathrm{E\varvec{D}})\theta= & {} K\sum _{k=1}^{K}\theta _{k}\sum _{l=1}^{K}\theta _{l}\sum _{m=1}^{M}\left( \int _{T_0}^{T_1}\left[ \left( \sum _{i=1}^{n-m}\frac{W_i^cW_{i+m}^c}{n-m}\right) ^2-(\textrm{E}W_i^cW_{i+m}^c)^2\right] B_kB_l\right) \nonumber \\= & {} o_p(1). \end{aligned}$$

(23)

Lemma 5 follows from Eqs. (20)–(23). $\square$

Proof of Theorem 1

In this proof, let C denote some positive constant not depending on n, but which may assume different values at each appearance. First, By Corollary 6.21 of Schumaker (2007), there exist a constant C such that

$$\begin{aligned} \beta _1(t)=\sum _{k=1}^{K}{\tilde{b}}_kB_k(t)+R(t) \end{aligned}$$

with

$$\begin{aligned} \sup _{t \in [T_0,T_1]} |R(t)| \le C K^{-r}. \end{aligned}$$

(24)

Let ${\tilde{\varvec{b}}}=({\tilde{b}}_1,\dots ,{\tilde{b}}_{K})^T$ and ${\tilde{\beta }}_1(t)=\sum _{k=1}^{K}{\tilde{b}}_kB_k(t)$. Thus,

$$\begin{aligned} \Vert {\hat{\beta }}_1-\beta _1\Vert ^2 \le 2\Vert {\hat{\beta }}_1-{\tilde{\beta }}_1\Vert ^2+ 2\Vert {\tilde{\beta }}_1-\beta _1\Vert ^2 \le 2\Vert {\hat{\beta }}_1-{\tilde{\beta }}_1\Vert ^2+C K^{-2r} \end{aligned}$$

(25)

Let $K\int _{T_0}^{T_1}{\hat{\varvec{G}}}(t)^T{\hat{\varvec{G}}}(t)\mathrm{{d}}t=\varvec{D}$. By Lemmas 1, 5 and Cauchy–Schwarz Inequality, we get

$$\begin{aligned} \Vert {\hat{\beta }}_1-{\tilde{\beta }}_1\Vert ^2= & {} \left\| \varvec{B}^T\left[ \left( \int _{T_0}^{T_1}{\hat{\varvec{G}}}(t)^T{\hat{\varvec{G}}}(t)\mathrm{{d}}t\right) ^{-1}\int _{T_0}^{T_1} {\hat{\varvec{G}}}(t)^T{\hat{\varvec{H}}}(t)\mathrm{{d}}t-{\tilde{\varvec{b}}}\right] \right\| ^2 \nonumber \\\le & {} C\frac{1}{K}\left\| \left( \int _{T_0}^{T_1}{\hat{\varvec{G}}}(t)^T{\hat{\varvec{G}}}(t)\mathrm{{d}}t\right) ^{-1}\int _{T_0}^{T_1} {\hat{\varvec{G}}}(t)^T\left( {\hat{\varvec{H}}}(t)-{\hat{\varvec{G}}}(t){\tilde{\varvec{b}}}\right) \mathrm{{d}}t\right\| ^2 \nonumber \\\le & {} CK\left\| \int _{T_0}^{T_1}{\hat{\varvec{G}}}(t)^T\left( {\hat{\varvec{H}}}(t)-{\hat{\varvec{G}}}(t){\tilde{\varvec{b}}}\right) \mathrm{{d}}t\right\| ^2 \nonumber \\\le & {} CK\sum _{m=1}^{M}\left[ \int _{T_0}^{T_1}\sum _{k=1}^{K}\sum _{i=1}^{n-m}\frac{B_k W_i^cW_{i+m}^c}{(n-m)^2}\sum _{j=1}^{n-m} \left( Y_j^cW_{j+m}^c-W_j^cW_{j+m}^c{\tilde{\beta }}_1\right) \right] ^2\nonumber \\\le & {} \frac{CK}{n-M}\sum _{m=1}^{M}AE, \end{aligned}$$

(26)

where

$$\begin{aligned} A=\int _{T_0}^{T_1}\left( \sum _{k=1}^{K}B_k\sum _{i=1}^{n-m}\frac{ W_i^cW_{i+m}^c}{n-m}\right) ^2, E=\int _{T_0}^{T_1}\left[ \sum _{j=1}^{n-m}\frac{Y_j^cW_{j+m}^c-W_j^cW_{j+m}^c{\tilde{\beta }}_1}{\sqrt{n-m}}\right] ^2. \end{aligned}$$

By Condition C3, Lemmas 1 and 4, we deduce that

$$\begin{aligned} \textrm{E}(A)\le & {} 2\int _{T_0}^{T_1}\left( \sum _{k=1}^{K}B_k\right) ^2\nonumber \\ {}{} & {} \textrm{E}\left[ \left( \frac{1}{n-m}\sum _{i=1}^{n-m}\frac{W_i^cW_{i+m}^c- \textrm{E}\left( W_i^c(t)W_{i+m}^c(t)\right) }{\sqrt{n-m}}\right) ^2+\left( \textrm{E}W_i^c(t)W_{i+m}^c(t)\right) ^2\right] \nonumber \\\le & {} 2C \int _{T_0}^{T_1}\left( \sum _{k=1}^{K}B_k\right) ^2 \le C \end{aligned}$$

(27)

Let

$$\begin{aligned} E_1=\textrm{E}\left\| \sum _{j=1}^{n-m}\frac{\left( Y_j^cW_{j+m}^c-\textrm{E}Y_j^cW_{j+m}^c\right) }{\sqrt{n-m}}\right\| ^2, E_2=\textrm{E}\left\| \sum _{j=1}^{n-m}\frac{\left( \textrm{E}Y_j^cW_{j+m}^c-\textrm{E}W_j^cW_{j+m}^c\beta _1\right) }{\sqrt{n-m}}\right\| ^2,\\ E_3=\textrm{E}\left\| \sum _{j=1}^{n-m}\frac{\left( \textrm{E}W_j^cW_{j+m}^c(\beta _1-{\tilde{\beta }}_1)\right) }{\sqrt{n-m}}\right\| ^2, E_4=\textrm{E}\left\| \sum _{j=1}^{n-m}\frac{\left( W_j^cW_{j+m}^c-\textrm{E}W_j^cW_{j+m}^c\right) {\tilde{\beta }}_1}{\sqrt{n-m}}\right\| ^2. \end{aligned}$$

Note that

$$\begin{aligned} E_1\le & {} 3\textrm{E}\left\| \sum _{j=1}^{n-m}\frac{\varepsilon _jW_{j+m}^c}{\sqrt{n-m}}\right\| ^2 + 3\textrm{E}\left\| \sum _{j=1}^{n-m}\frac{e_j^cW_{j+m}^c}{\sqrt{n-m}}\right\| ^2\nonumber \\{} & {} + 3\textrm{E}\left\| \sum _{j=1}^{n-m}\frac{\left( W_j^cW_{j+m}^c-\textrm{E}W_j^cW_{j+m}^c\right) \beta _1}{\sqrt{n-m}}\right\| ^2\nonumber \\= & {} 3E_{11}+3E_{12}+3E_{13}. \end{aligned}$$

(28)

By Conditions C4, we deduce that there exists a constant $C_0$ such that $\sup _{t \in [T_0,T_1]}| \beta _1(t)|<C_0$. Hence, using Lemma 4 and Conditions C1–C2, we have

$$\begin{aligned} E_{11}\le \frac{3}{n-m}\sum _{j=1}^{n-m}\textrm{E}\left\| \varepsilon _j^2\right\| \textrm{E}\left\| (W_{j+m}^c)^2\right\| <C, \end{aligned}$$

(29)

$$\begin{aligned} E_{12}\le \frac{3}{n-m}\sum _{j=1}^{n-m}\textrm{E}\left\| (e_j^c)^2\right\| \textrm{E}\left\| (W_{j+m}^c)^2\right\| <C \end{aligned}$$

(30)

and

$$\begin{aligned} E_{13}\le \frac{3C_0}{n-m}\sum _{j=1}^{n-m}\Vert \left( W_j^cW_{j+m}^c-\textrm{E}W_j^cW_{j+m}^c\right) \Vert ^2<C. \end{aligned}$$

(31)

Combining Eqs. (28)–(31),we deduce that

$$\begin{aligned} E_1\le C. \end{aligned}$$

(32)

Seen from Eq. (4), we have

$$\begin{aligned} E_2=0. \end{aligned}$$

(33)

By using Lemma 5 and Eq. (24), we get

$$\begin{aligned} E_3\le \sup _{t \in [T_0,T_1]} |R(t)|^2 \textrm{E}\left\| W_i^cW_{i+m}^c\right\| ^2 \le C K^{-2r}\le C. \end{aligned}$$

(34)

$$\begin{aligned} E_4\le \sup _{t \in [T_0,T_1]} |{\tilde{\beta }}_1(t)|^2 \textrm{E}\left\| \frac{1}{\sqrt{n-m}}\sum _{i=1}^{n-m}\left[ W_i^cW_{i+m}^c-\textrm{E}\left( W_i^cW_{i+m}^c\right) \right] \right\| ^2 \le C. \end{aligned}$$

(35)

Combining Eqs. (32)–(35), we deduce that

$$\begin{aligned} \textrm{E}(E)\le 4E_1+4E_2+4E_3+4E_4\le C. \end{aligned}$$

(36)

Hence, by combining Eqs. (26), (27) and (37), we have

$$\begin{aligned} \Vert {\hat{\beta }}_1-{\tilde{\beta }}_1\Vert ^2 =O_p\left( n^{-1}K\right) . \end{aligned}$$

(37)

Thus, Eq. (13) follows from Eqs. (25) and (37). This completes the proof of Theorem 1. $\square$

Proof of Theorem 2

In this proof, let C denote some positive constant not depending on n, but which may assume different values at each appearance. By using Theorem 2.17 of Bosq (2012), we have

$$\begin{aligned} \textrm{E}\left\| \frac{1}{\sqrt{n}}\sum _{i=1}^{n}\left( W_i-\textrm{E}W\right) \right\| ^2<C. \end{aligned}$$

According to condition C2, we deduce that

$$\begin{aligned} \sup _{t \in [T_0,T_1]}\left( \frac{1}{n}\sum _{i=1}^{n}W_i(t)\right) ^2\le & {} \frac{2}{n}\sup _{t \in [T_0,T_1]}\left( \sum _{i=1}^{n}\frac{W_i(t) -\textrm{E}W(t)}{\sqrt{n}}\right) ^2\nonumber \\{} & {} +2\sup _{t \in [T_0,T_1]}\left( \textrm{E}W(t)\right) ^2=O_p(1). \end{aligned}$$

(38)

Since $e_i(t)(i=1,\dots ,n)$ and $\varepsilon _i(t)(i=1,\dots ,n)$ are independent identically distribution zero mean random process, we have

$$\begin{aligned} \textrm{E}\left\| \frac{1}{\sqrt{n}}\sum _{i=1}^{n}e_i\right\| ^2<C. \end{aligned}$$

(39)

and

$$\begin{aligned} \textrm{E}\left\| \frac{1}{\sqrt{n}}\sum _{i=1}^{n}\varepsilon _i\right\| ^2<C. \end{aligned}$$

(40)

According to condition C4 and Eq. (39), we have

$$\begin{aligned} \left\| \beta _1\frac{1}{n}\sum _{i=1}^{n}e_i\right\| ^2 \le \sup _{t \in [T_0,T_1]}|\beta _1(t)|^2\Vert \frac{1}{n}\sum _{i=1}^{n}e_i\Vert ^2=O_p\left( n^{-1}\right) . \end{aligned}$$

(41)

Combining Eqs. (1), (12), (38), (40), (41) and Theorem 1, we have

$$\begin{aligned} \Vert {\hat{\beta }}_0-\beta _0\Vert ^2\le & {} 3\left\| \frac{1}{n}\sum _{i=1}^{n}W_i\left( {\hat{\beta }}_1-\beta _1\right) \right\| ^2+ 3\left\| \beta _1\frac{1}{n}\sum _{i=1}^{n}e_i\right\| ^2 +3\left\| \frac{1}{n}\sum _{i=1}^{n}\varepsilon _i\right\| ^2\\\le & {} 3\sup _{t \in [T_0,T_1]}\left( \frac{1}{n}\sum _{i=1}^{n}W_i(t)\right) ^2\Vert {\hat{\beta }}_1-\beta _1\Vert ^2+O_p\left( n^{-1}\right) +O_p\left( n^{-1}\right) \\= & {} O_p\left( n^{-1}K+K^{-2r}\right) \end{aligned}$$

This completes the proof of Theorem 2. $\square$