Estimation for functional linear semiparametric model

Abstract

We study a functional linear semiparametric model which is not only an extension of partially functional linear models, but also an extension of semiparametric models. We consider the case that a response is related to a functional predictor and several scalar variables and the functional predictor is observed at a set of discrete points with noise. We propose a new estimation procedure which combines functional principal component analysis and B-spline methods to estimate unknown parameters and functions in model. The asymptotic distribution of the estimators of slope parameters is derived and the global convergence rate of the estimator of unknown slope function is established. The convergence rate of the mean squared prediction error for a predictor is also established. Simulation studies are conducted to investigate the finite sample performance of the proposed estimators. A real data example based on real estate data is used to illustrate our proposed methodology.

Appendix: Proofs of theorems

Let $C>0$ denote a generic constant of which the value may change from line to line. For a matrix $\pmb {A}=(a_{ij})$, set $\Vert \pmb {A}\Vert _{\infty }=\max _{i}\sum _{j}|a_{ij}|$ and $|\pmb {A}|_{\infty }=\max _{i,j}|a_{ij}|$. For a vector $\pmb {v}=(v_{1},\ldots ,v_{k})^{T}$, set $\Vert \pmb {v}\Vert _{\infty }=\sum _{j=1}^{k}|v_{j}|$ and $|\pmb {v}|_{\infty }=\max _{1\le j\le k}|v_{j}|$.

Denote $A_{l}=\sum _{j=1}^{\infty }a_{j}\xi _{lj}$, ${\tilde{A}}_{i}=A_{i}-\frac{1}{n}\sum _{l=1}^{n}A_{l}{\tilde{\zeta }}_{li}$, $F_{i}=f(U_{i})$, ${\tilde{F}}_{i}=F_{i}-\frac{1}{n}\sum _{l=1}^{n}F_{l}{\tilde{\zeta }}_{li}$, ${\tilde{\varepsilon }}_{i}=\varepsilon _{i}-\frac{1}{n}\sum _{l=1}^{n}\varepsilon _{l}{\tilde{\zeta }}_{li}$ and $\tilde{\pmb {A}}=({\tilde{A}}_{1},\ldots ,{\tilde{A}}_{n})^{T}$, $\tilde{\pmb {F}}=({\tilde{F}}_{1},\ldots ,{\tilde{F}}_{n})^{T}$, $\tilde{\pmb {\varepsilon }}=({\tilde{\varepsilon }}_{1},\ldots ,{\tilde{\varepsilon }}_{n})^{T}$.

We first list the following Lemmas A.1–A.8, their proofs are given in supplementary material.

Lemma A.1

Let $\Delta (s,t)={\hat{K}}(s,t)-K(s,t)$ and $|\Vert \Delta \Vert |=(\int _{{\mathcal {T}}}\int _{{\mathcal {T}}}\Delta ^{2}(s,t)dsdt)^{1/2}$. suppose that Assumptions 1–3 and 6 hold, then it holds that

$$\begin{aligned} |\Vert \Delta \Vert |=O_{p}(n^{-1/2}). \end{aligned}$$

Lemma A.2

Suppose that Assumptions 1–3 and 6 hold, then it holds that

$$\begin{aligned} \frac{1}{n}\tilde{\pmb {Z}}^{T}\tilde{\pmb {Z}}=E(\pmb {Z}^{*}\pmb {Z}^{*T})+o_{p}(1). \end{aligned}$$

Lemma A.3

Assume that Assumptions 1–6 hold. Then it holds that

$$\begin{aligned} \frac{1}{n}\sum _{i=1}^{n}{\tilde{B}}_{k}(U_{i}){\tilde{B}}_{k'}(U_{i})=E(B_{k}(U)B_{k'}(U))+o_{p}(h_{0}^{2}), \end{aligned}$$

where $o_{p}(h_{0}^{2})$ holds uniformly for $1\le k,k^{\prime }\le K_{n}$.

Lemma A.4

Under Assumptions 1–4, it holds that

$$\begin{aligned} \sum _{j=1}^{m}\lambda _{j}\left[ a_{j}-\frac{1}{{\hat{\lambda }}_{j}}\left( \frac{1}{n}\sum _{l=1}^{n}A_{l}{\hat{\xi }}_{lj}\right) \right] ^{2}=O_{p}(n^{-1}\lambda _{m}^{-1}m). \end{aligned}$$

Lemma A.5

Under Assumptions 1–3, it holds that

$$\begin{aligned} \sum _{j=1}^{m}\lambda _{j}^{-1}\left( \sum _{i=1}^{n}\xi _{ij}{\tilde{Z}}_{ik}\right) ^{2}=O_{p}(n\lambda _{m}^{-1}m^{2}+nm^{4}\log m). \end{aligned}$$

Lemma A.6

Under the Assumptions 1–4 and 6, it holds that

$$\begin{aligned} n^{-1/2}\left| \sum _{j=1}^{m}\frac{1}{{\hat{\lambda }}_{j}}\left( \frac{1}{n}\sum _{l=1}^{n}A_{l}{\hat{\xi }}_{lj}\right) \sum _{i=1}^{n}({\hat{\xi }}_{ij}-\xi _{ij}){\tilde{Z}}_{ik}\right| =o_{p}(1). \end{aligned}$$

Lemma A.7

Under the Assumptions 1–4 and 6, it holds that

$$\begin{aligned} n^{-1/2}\left| \sum _{i=1}^{n}{\tilde{A}}_{i}{\tilde{Z}}_{ik}\right| =o_{p}(1). \end{aligned}$$

Lemma A.8

Define ${\check{a}}_{j}=\frac{1}{{\hat{\lambda }}_{j} }E[(Y-Z^{T}\beta _{0}-f(U))\xi _{j}]$. Under the assumptions of Theorem 3.2, it holds that

$$\begin{aligned} \sum _{j=1}^{m}\left( {\hat{a}}_{j}-{\check{a}}_{j})^{2}=O_{p}(n^{-1}m\lambda _{m}^{-1} +n^{-2}m\lambda _{m}^{-2}\sum _{j=1}^{m}a_{j}^{2}\lambda _{j}^{-2}j^{3}+m\lambda _{m}^{-2}\left( h_{N}^{2r}+\frac{1}{Nh_{N}}\right) \right) . \end{aligned}$$

Proof of Theorem 3.1

Under Assumption 5, according to Corollary 6.21 of (Schumaker 1981, p.227), there exists a spline function $f_{0}(u)=\sum _{k=1}^{K_{n}}b_{0k}B_{k}(u)$ and a constant $C>0$ such that

$$\begin{aligned} \sup _{u\in [U_{0},U^{0}]}|{\bar{f}}(u)|\le Ch_{0}^{\rho '}, \end{aligned}$$

(6.1)

where ${\bar{f}}(u)=f(u)-f_{0}(u)$. Denote ${\bar{F}}_{i}={\bar{f}}(U_{i})$, $\tilde{{\bar{F}}}_{i}={\bar{F}}_{i}-\frac{1}{n}\sum _{l=1}^{n}{\bar{F}}_{l}{\tilde{\xi }}_{li}$, $\tilde{\bar{\pmb {F}}}=(\tilde{{\bar{F}}}_{1},\ldots ,\tilde{{\bar{F}}}_{n})^{T}$. By (2.14), we have

$$\begin{aligned} \hat{\pmb {\beta }}-\pmb {\beta }_{0}=(\tilde{\pmb {Z}}^{T}\tilde{\pmb {Z}}-\tilde{\pmb {Z}}^{T}\tilde{\pmb {B}}(\tilde{\pmb {B}}^{T}\tilde{\pmb {B}})^{-1}\tilde{\pmb {B}}^{T}\tilde{\pmb {Z}})^{-1} \tilde{\pmb {Z}}^{T}(\pmb {I}_{n}-\tilde{\pmb {B}}(\tilde{\pmb {B}}^{T}\tilde{\pmb {B}})^{-1}\tilde{\pmb {B}}^{T})(\tilde{\pmb {A}}+\tilde{\bar{\pmb {F}}}+\tilde{\pmb {\varepsilon }}), \nonumber \\\end{aligned}$$

(6.2)

where $\pmb {I}_{n}$ is the $n\times n$ identity matrix. By arguments similar to those used in the proof of Lemma 1 of Tang (2013) and using Lemmas A.2 and A.3, we have

$$\begin{aligned} \frac{1}{n}(\tilde{\pmb {Z}}^{T}\tilde{\pmb {Z}}-\tilde{\pmb {Z}}^{T}\tilde{\pmb {B}}(\tilde{\pmb {B}}^{T}\tilde{\pmb {B}})^{-1}\tilde{\pmb {B}}^{T}\tilde{\pmb {Z}})=\pmb {\Sigma }_{n}+o_{p}(1). \end{aligned}$$

(6.3)

Similar to the proof of Lemma A.7, we have that $n^{-1/2}|\sum _{i=1}^{n}{\tilde{A}}_{i}{\tilde{B}}_{k}(U_{i})|=o_{p}(h_{0})$ uniformly for $1\le k\le K_{n}$. Hence by arguments similar to those used in the proof of Lemma 1 of Tang (2013), we obtain that

$$\begin{aligned} \begin{array}{ll} n^{-\frac{1}{2}}|\tilde{\pmb {Z}}^{T}\tilde{\pmb {B}}(\tilde{\pmb {B}}^{T}\tilde{\pmb {B}})^{-1}\tilde{\pmb {B}}^{T}\tilde{\pmb {A}}|_{\infty } &{} \le K_{n}\Vert \frac{1}{n}\tilde{\pmb {Z}}^{T}\tilde{\pmb {B}}\Vert _{\infty }\Vert (\frac{K_{n}}{n}\tilde{\pmb {B}}^{T}\tilde{\pmb {B}})^{-1}\Vert _{\infty }|n^{-\frac{1}{2}}\tilde{\pmb {B}}^{T}\tilde{\pmb {A}}|_{\infty } \\ &{} =K_{n}O_{p}(1)O_{p}(1)o_{p}(h_{0})=o_{p}(1) \end{array} \nonumber \\\end{aligned}$$

(6.4)

Using Lemma A.7 and (6.4), we deduce that

$$\begin{aligned} n^{-\frac{1}{2}}\tilde{\pmb {Z}}^{T}(\pmb {I}_{n}-\tilde{\pmb {B}}(\tilde{\pmb {B}}^{T}\tilde{\pmb {B}})^{-1}\tilde{\pmb {B}}^{T})\tilde{\pmb {A}}=o_{p}(1) \end{aligned}$$

(6.5)

By Lemma A.2, we get that $\sum _{i=1}^{n}{\tilde{Z}}_{ik}^{2}=O_{p}(n)$. Using (6.1) and the assumption that $nh_{0}^{2\rho '}\rightarrow 0$, we have

$$\begin{aligned} n^{-1}\left( \sum _{i=1}^{n}{\tilde{Z}}_{ik}{\bar{F}}_{i}\right) ^{2}\le n^{-1}\left( \sum _{i=1}^{n}{\bar{F}}_{i}^{2}\right) \left( \sum _{i=1}^{n}{\tilde{Z}}_{ik}^{2}\right) =o_{p}(1). \end{aligned}$$

By arguments similar to those used to prove Lemmas A.5 and A.6 and using (6.1), we deduce that $n^{-\frac{1}{2}}\sum _{i=1}^{n}\left( \frac{1}{n}\sum _{l=1}^{n}{\bar{F}}_{l}{\tilde{\xi }}_{li}\right) {\tilde{Z}}_{ik}=o_{p}(1)$ and

$$\begin{aligned} n^{-\frac{1}{2}}\sum _{i=1}^{n}(\frac{1}{n}\sum _{l=1}^{n}\varepsilon _{l}{\tilde{\xi }}_{li}){\tilde{Z}}_{ik}=o_{p}(1). \end{aligned}$$

(6.6)

Hence $n^{-\frac{1}{2}}\sum _{i=1}^{n}{\tilde{Z}}_{ik}\tilde{{\bar{F}}}_{i}=o_{p}(1)$. By arguments similar to those used to prove (6.5), we further get that

$$\begin{aligned} n^{-\frac{1}{2}}\tilde{\pmb {Z}}^{T}(\pmb {I}_{n}-\tilde{\pmb {B}}(\tilde{\pmb {B}}^{T}\tilde{\pmb {B}})^{-1}\tilde{\pmb {B}}^{T})\tilde{\bar{\pmb {F}}}=o_{p}(1) \end{aligned}$$

(6.7)

We decompose $\sum _{i=1}^{n}{\tilde{Z}}_{ik}\varepsilon _{i}$ into three terms as

$$\begin{aligned} \sum _{i=1}^{n}{\tilde{Z}}_{ik}\varepsilon _{i}= & {} \sum _{i=1}^{n}\varepsilon _{i}\left( Z_{ik}-\sum _{j=1}^{m}\frac{E(Z_{lk}\xi _{j})}{\lambda _{j}}\xi _{ij}\right) \\&-\,\sum _{i=1}^{n}\varepsilon _{i}\sum _{j=1}^{m}\frac{\xi _{ij}}{\lambda _{j}}\left( \frac{1}{n}\sum _{l=1}^{n}Z_{lk}\xi _{lj}-E(Z_{lk}\xi _{j})\right) \\&-\,\sum _{i=1}^{n}\varepsilon _{i}\frac{1}{n}\sum _{l=1}^{n}Z_{lk}(\tilde{\zeta }_{li}-\vec {\zeta }_{li}). \end{aligned}$$

Similar to the proof of Lemma A.6, we have $\sum _{i=1}^{n}\varepsilon _{i} \frac{1}{n}\sum _{l=1}^{n}Z_{lk}({\tilde{\zeta }}_{li}-\vec {\zeta }_{li})=o_{p}(n)$. Since

$$\begin{aligned} \sum _{i=1}^{n}\varepsilon _{i}\left( Z_{ik}-\sum _{j=1}^{m}\frac{E(Z_{lk}\xi _{j} )}{\lambda _{j}}\xi _{ij}\right) =\sum _{i=1}^{n}\varepsilon _{i}Z_{ik}^{*}+\sum _{i=1} ^{n}\varepsilon _{i}\sum _{j=m+1}^{\infty }\mu _{kj}\xi _{ij}, \end{aligned}$$

$\sum _{i=1}^{n}\varepsilon _{i}\sum _{j=1}^{m}\frac{\xi _{ij}}{\lambda _{j} }(\frac{1}{n}\sum _{l=1}^{n}Z_{lk}\xi _{lj}-E(Z_{lk}\xi _{j}))=o_{p}(n)$ and $\sum _{i=1}^{n}\varepsilon _{i}\sum _{j=m+1}^{\infty }\mu _{kj}\xi _{ij} =o_{p}(n)$, then it follows by (6.6) that

$$\begin{aligned} n^{-\frac{1}{2}}\sum _{i=1}^{n}{\tilde{Z}}_{ik}{\tilde{\varepsilon }}_{i}=n^{-\frac{1}{2}}\sum _{i=1}^{n}Z_{ik}^{*}\varepsilon _{i}+o_{p}(1). \end{aligned}$$

(6.8)

By arguments similar to those used in the proof of Lemma 2 of Tang (2013), we deduce that

$$\begin{aligned} n^{-\frac{1}{2}}\tilde{\pmb {Z}}^{T}\tilde{\pmb {B}}(\tilde{\pmb {B}}^{T}\tilde{\pmb {B}})^{-1}\tilde{\pmb {B}}^{T}\tilde{\pmb {\varepsilon }} =n^{-\frac{1}{2}}\pmb {\Upsilon }_{n}\pmb {\Pi }_{n}^{-1}\pmb {B}^{T}\pmb {\varepsilon }+o_{p}(1) , \end{aligned}$$

(6.9)

where $\pmb {B}^{T}=(\pmb {B}_{1},\ldots ,\pmb {B}_{n})$ with $\pmb {B}_{i}=(B_{1}(U_{i}), \ldots ,B_{K_{n}}(U_{i}))^{T}$ and $\pmb {\varepsilon }=(\varepsilon _{1},\ldots ,\varepsilon _{n})^{T}$. Now (3.1) follows from (6.2), (6.3), (6.4), (6.6)–(6.9) and the central limit theorem. The proof of Theorem 3.1 is finished.

Proof of Theorem 3.2

Note that

$$\begin{aligned}&\int _{{\mathcal {T}}}[{\hat{a}}(t)-a(t)]^{2}dt\nonumber \\&\quad \le C\left( \sum _{j=1}^{m}({\hat{a}}_{j}-{\check{a}}_{j})^{2}+\sum _{j=1}^{m}({\check{a}}_{j}-a_{j})^{2} +m\sum _{j=1}^{m}a_{j}^{2}\Vert {\hat{\phi }}_{j}-\phi _{j}\Vert ^{2} +\sum _{j=m+1}^{\infty }a_{j}^{2}\right) \nonumber \\\end{aligned}$$

(5.10)

and

$$\begin{aligned} \begin{array}{ll} \sum _{j=1}^{m}({\check{a}}_{j}-a_{j})^{2}=\sum _{j=1}^{m}\frac{({\hat{\lambda }}_{j}-\lambda _{j})^{2}}{\lambda _{j}^{2}}a_{j}^{2}[1+o_{p}(1)] =O_{p}(n^{-1}\lambda _{m}^{-1}\sum _{j=1}^{m}a_{j}^{2}\lambda _{j}^{-1}). \end{array}\nonumber \\\end{aligned}$$

(5.11)

Assumption 4 implies that $m\sum _{j=1}^{m}a_{j}^{2}\Vert {\hat{\phi }}_{j}-\phi _{j}\Vert ^{2}=O_{p}(mn^{-1}\sum _{j=1}^{m}a_{j}^{2}j^{2}\log j)=o_{p}(m/n)$ and $\sum _{j=m+1}^{\infty }a_{j}^{2}=O(m^{-2\gamma +1})$. Now (3.4) follows from Lemma A.8, (5.10) and (5.11). The proof of Theorem 3.2 is finished.

Proof of Theorem 3.3

By Assumption 7 and Lemma A.3, all the eigenvalues of $(\frac{K^{*}_{n}}{n}\tilde{\pmb {B}}^{*T}\tilde{\pmb {B}}^{*})^{-1}$ are bounded away from zero and infinity, except possibly on an event whose probability tends to zero. Similar to (6.1), there exists a spline function $f^{*}(u)=\sum _{k=1}^{K^{*} _{n}}b_{0k}^{*}B_{k}^{*}(u)$ such that

$$\begin{aligned} \sup _{u\in [U_{0},U^{0}]}|f(u)-f^{*}(u)|\le Ch^{\rho '}. \end{aligned}$$

(5.12)

Let $\pmb {b}^{*}_{0}=(b^{*}_{01},\ldots ,b^{*}_{0K^{*}_{n}})^{T}$. Using the properties of B-splines (de Boor 1978), we obtain

$$\begin{aligned} \begin{array} [c]{ll} \int _{U_{0}}^{U^{0}}({\hat{f}}(u)-f(u))^{2}du &{} \le C(\Vert \hat{\pmb {b}}-\pmb {b}^{*} _{0}\Vert ^{2}/K^{*}_{n}+h^{2\rho '}).\\ &{} \end{array} \end{aligned}$$

(5.13)

By arguments similar to those used to prove (6.4)–(6.7) and using Theorem 3.1, we conclude that $\Vert \hat{\pmb {b}}-\pmb {b}^{*} _{0}\Vert ^{2}=O_{p}(n^{-1}{K^{*}_{n}}^{2})$. Now (3.5) follows from (5.13) and the fact that $h=O({K^{*}_{n}}^{-1})$. This completes the proof of Theorem 3.3.

Proof of Theorem 3.4

Observe that

$$\begin{aligned} \begin{array} [c]{ll} \text{ MSPE } &{} \le 2\{\Vert {\hat{a}}\Vert ^{2}\cdot \Vert {\hat{X}}_{n+1}-X_{n+1}\Vert ^{2}+\Vert {\hat{a}}-a\Vert _{K}^{2}+({\hat{\beta }} -\beta _{0})^{T}E(\pmb {Z}\pmb {Z}^{T}) \\ &{}\quad \times \,(\hat{\pmb {\beta }}-\pmb {\beta }_{0})+E([{\hat{f}}(U_{n+1})-f(U_{n+1})]^{2}|{\mathcal {S}})\}, \end{array} \end{aligned}$$

(5.14)

where $\Vert {\hat{a}}-a\Vert _{K}^{2}=\int _{{\mathcal {T}}}\int _{{\mathcal {T}} }K(s,t)[{\hat{a}}(s)-a(s)][{\hat{a}}(t)-a(t)]dsdt$. Similar to the proof of Lemma A.1, we obtain that

$$\begin{aligned} \Vert {\hat{X}}_{n+1}-X_{n+1}\Vert ^{2}=O_{p}\left( h_{N}^{2r}+\frac{1}{N_{n+1}h_{N}}\right) . \end{aligned}$$

(5.15)

Under the assumptions of Theorem 3.4, using arguments similar to those used in the proof of Theorem 2 of Tang (2015), we deduce that

$$\begin{aligned} \Vert {\hat{a}}-a\Vert _{K}^{2}=O_{p}(n^{-(\delta +2\gamma -1)/(\delta +2\gamma )}). \end{aligned}$$

(5.16)

Write

$$\begin{aligned} {\hat{f}}(U_{n+1})-g(U_{n+1})=\hat{f}(U_{n+1})-f^{*}(U_{n+1})+f^{*}(U_{n+1})-f(U_{n+1}). \end{aligned}$$

Using Theorem 3.3, we obtain $E([{\hat{f}}(U_{n+1})-f^{*}(U_{n+1})]^{2}|{\mathcal {S}}) =O_{p}(n^{-2\rho '/(2\rho '+1)})$. Using (6.12), we obtain $E([f^{*}(U_{n+1})-f(U_{n+1}]^{2}|{\mathcal {S}})=O_{p}(h^{2\rho '})$. Hence, $E([{\hat{f}}(U_{n+1})-f(U_{n+1})]^{2}|{\mathcal {S}})=O_{p}(n^{-2\rho '/(2\rho '+1)})$. Now (3.7) follows from (5.14)–(5.16), Assumption 6 and Theorem 3.1. This completes the proof of Theorem 3.4.