A smooth simultaneous confidence band for correlation curve

Abstract

A plug-in estimator is proposed for a local measure of variance explained by regression, termed correlation curve in Doksum et al. (J Am Stat Assoc 89:571–582, 1994), consisting of a two-step spline–kernel estimator of the conditional variance function and local quadratic estimator of first derivative of the mean function. The estimator is oracally efficient in the sense that it is as efficient as an infeasible correlation estimator with the variance function known. As a consequence of the oracle efficiency, a smooth simultaneous confidence band (SCB) is constructed around the proposed correlation curve estimator and shown to be asymptotically correct. Simulated examples illustrate the versatility of the proposed oracle SCB which confirms the asymptotic theory. Application to a 1995 British Family Expenditure Survey data has found marginally significant evidence for a local version of Engel’s law, i.e., food budget share and household real income are inversely related (Hamilton in Am Econ Rev 91:619–630, 2001).

Appendix

Throughout this section, for any function $g\left( u\right)$, define $\left\| g\right\| _{\infty }=\sup \nolimits _{u\in \mathcal {I} _{n}}\left| g\left( u\right) \right|$. For any vector $\xi$, one denotes by $\left\| \xi \right\|$ the Euclidean norm and $\left\| \xi \right\| _{\infty }$ means the largest absolute value of the elements. We use C to denote any positive constants in the generic sense. A random sequence $\left\{ X_{n}\right\}$ “bounded in probability” is denoted as $X_{n}=\mathcal {O}_{p}\left( 1\right)$, while $X_{n}=o_{p}\left( 1\right)$ denotes convergence to 0 in probability. A sequence of random functions which are $o_{p}$ or $\mathcal {O}_{p}$ uniformly over $x\in \mathcal {I}_{n}$ denoted as $u_{p}$ or $U_{p}$.

Next, we state the strong approximation Theorem of Tusnády (1977). It will be used later in the proof of Lemmas 4 and 5.

Let $U_{1},\ldots ,U_{n}$ be i.i.d r.v.’s on the 2-dimensional unit square with $\mathbb {P}\left( U_{i}<\mathbf {t}\right) =\lambda \left( \mathbf {t} \right) ,\mathbf {0}\le \mathbf {t}\le \mathbf {1}$, where $\mathbf {t}=$ $\left( t_{1},t_{2}\right)$ and $\mathbf {1}=\left( 1,1\right)$ are 2-dimensional vectors, $\lambda \left( \mathbf {t}\right) =t_{1}t_{2}$. The empirical distribution function $F_{n}^{u}\left( \mathbf {t}\right) =n^{-1}\sum _{i=1}^{n}I_{\left\{ U_{i}<\mathbf {t}\right\} }$ for $\mathbf {0}\le \mathbf {t}\le \mathbf {1}$.

Lemma 1

The 2-dimensional Brownian bridge $B\left( \mathbf {t}\right)$ is defined by $B\left( \mathbf {t}\right) =W\left( \mathbf {t}\right) -\lambda \left( \mathbf {t}\right) W\left( \mathbf {1} \right)$ for $\mathbf {0\le t\le 1}$, where $W\left( \mathbf {t}\right)$ is a 2-dimensional Wiener process. Then there is a version $B_{n}\left( \mathbf {t}\right)$ of $B\left( \mathbf {t}\right)$ such that

$$\begin{aligned} \mathbb {P}\left[ \sup _{\mathbf {0}\le \mathbf {t}\le \mathbf {1}}\left| n^{1/2}\left\{ F_{n}^{u}\left( \mathbf {t}\right) -\lambda \left( \mathbf {t} \right) \right\} -B_{n}\left( \mathbf {t}\right) \right| >n^{-1/2}\left( C\log n+x\right) \log n\right] <Ke^{-\lambda x}, \end{aligned}$$

holds for all x, where $C, K, \lambda$ are positive constants.

The Rosenblatt transformation for bivariate continuous $\left( X,\varepsilon \right)$ is

$$\begin{aligned} \left( X^{^{*}},\varepsilon ^{*}\right) =M\left( X,\varepsilon \right) =\left( F_{X}\left( x\right) ,F_{\varepsilon |X}\left( \varepsilon \left| x\right. \right) \right) , \end{aligned}$$

(29)

then $\left( X^{*},\varepsilon ^{*}\right)$ has uniform distribution on $\left[ a,b\right] ^{2}$; therefore,

$$\begin{aligned} Z_{n}\left\{ M^{-1}\left( x^{*},\varepsilon ^{*}\right) \right\} =Z_{n}\left( x,\varepsilon \right) =n^{1/2}\left\{ F_{n}\left( x,\varepsilon \right) -F\left( x,\varepsilon \right) \right\} , \end{aligned}$$

with $F_{n}\left( x,\varepsilon \right)$ denoting the empirical distribution of $\left( X,\varepsilon \right)$. Lemma 1 implies that there exists a version $B_{n}$ of 2 -dimensional Brownian bridge such that

$$\begin{aligned} \sup _{x,\varepsilon }\left| Z_{n}\left( x,\varepsilon \right) -B_{n}\left\{ M(x,\varepsilon )\right\} \right| =\mathcal {O} _{a.s.}\left( n^{-1/2}\log ^{2}n\right) . \end{aligned}$$

(30)

Lemma 2

Under Assumptions (A2) and (A5), there exists $\alpha _{1}>0$ such that the sequence $D_{n}=n^{\alpha _{1}}$ satisfies

$$\begin{aligned} n^{-1/2}h_{1}^{-1/2}D_{n}\log ^{2}n\rightarrow & {} 0,n^{1/2}h_{1}^{1/2}D_{n}^{-\left( 1+\eta _{1}\right) }\rightarrow 0, \\ \sum \nolimits _{n=1}^{\infty }D_{n}^{-\left( 2+\eta _{1}\right) }< & {} \infty ,D_{n}^{-\eta _{1}}h_{1}^{-1/2}\rightarrow 0. \end{aligned}$$

For such a sequence $\left\{ D_{n}\right\}$,

$$\begin{aligned} \mathbb {P}\left\{ \omega \shortmid \exists N\left( \omega \right) ,\left| \varepsilon _{i}\left( \omega \right) \right| <D_{n},1\le i\le n,n>N\left( \omega \right) \right\} =1. \end{aligned}$$

(31)

Lemma 3

Under Assumptions (A1)–(A5), as $n\rightarrow \infty$,

$$\begin{aligned} \tilde{\rho }_{\mathrm {LQ}}\left( x\right) -\rho (x)= & {} \sigma _{1}\left\{ 1-\rho ^{2}\left( x\right) \right\} ^{3/2}\sigma ^{-1}\left( x\right) \mu _{3}(K^{*})\beta ^{\prime \prime }\left( x\right) h_{1}^{2}/6 \nonumber \\&+\,\sigma _{1}\left\{ 1-\rho ^{2}\left( x\right) \right\} ^{3/2}\sigma ^{-1}\left( x\right) n^{-1}h_{1}^{-1}f^{-1}\left( x\right) \nonumber \\&\times \sum \nolimits _{i=1}^{n}K_{h_{1}}^{*}(X_{i}-x)\sigma \left( X_{i}\right) \varepsilon _{i} \nonumber \\&+\,u_{p}\left( h_{1}^{2}+n^{-1/2}h_{1}^{-3/2}\log ^{-1/2}n\right) . \end{aligned}$$

(32)

Proof

From the definition of $\tilde{\rho }_{\mathrm {LQ}}(x)$ in ( 15), the Taylor series expansions, and $\hat{\beta }\left( x\right) -\beta \left( x\right) =U_{p}\left( n^{-1/2}h_{1}^{-3/2}\log ^{1/2}n+h_{1}^{2}\right)$, one has

$$\begin{aligned} \tilde{\rho }_{\mathrm {LQ}}\left( x\right) -\rho (x)= & {} \sigma _{1}\left\{ 1-\rho ^{2}\left( x\right) \right\} ^{3/2}\sigma ^{-1}\left( x\right) \left\{ \hat{\beta }\left( x\right) -\beta \left( x\right) \right\} \\&+\,U_{p}\left( n^{-1}h_{1}^{-3}\log n+h_{1}^{4}\right) .\nonumber \end{aligned}$$

(33)

Write $\mathbf {Y}$ as the sum of a signal vector $\varvec{\mu =}\left\{ \mu \left( X_{1}\right) ,\ldots ,\mu \left( X_{n}\right) \right\} ^{\scriptstyle { T}}$ and a noise vector $\mathbf {E=}\left\{ \sigma \left( X_{1}\right) \varepsilon _{1},\ldots ,\sigma \left( X_{n}\right) \varepsilon _{n}\right\} ^{ \scriptstyle {T}}$,

$$\begin{aligned} \mathbf {Y}=\varvec{\mu }+\mathbf {E}. \end{aligned}$$

(34)

The local quadratic estimator $\hat{\beta }\left( x\right)$ has a noise and bias error decomposition

$$\begin{aligned} \hat{\beta }\left( x\right) -\beta \left( x\right) =I(x)+II(x), \end{aligned}$$

in which the bias term I(x) and noise term II(x) are

$$\begin{aligned} I(x)= & {} e_{1}^{^{\scriptstyle {T}}}(\mathbf {X}^{\scriptstyle {T}}\mathbf {WX})^{-1}\mathbf {X}^{\scriptstyle {T}}\mathbf {W} \nonumber \\&\times \, \left\{ \varvec{\mu }-\mu \left( x\right) \mathbf {X}e_{0}-\beta \left( x\right) \mathbf {X}e_{1}-\beta ^{\prime }\left( x\right) /2\mathbf {X} e_{2}\right\} , \end{aligned}$$

(35)

$$\begin{aligned} II(x)= & {} e_{1}^{^{\scriptstyle {T}}}(\mathbf {X}^{\scriptstyle {T}}\mathbf {WX} )^{-1}\mathbf {X}^{\scriptstyle {T}}\mathbf {WE.} \end{aligned}$$

(36)

where $e_{k}, k=0,1,2$, as defined in (8), $\mathbf {X}$ in (9), $\mathbf {W}$ in (10), $\varvec{\mu }$ and $\mathbf {E}$ in (34). Standard arguments from kernel smoothing theory yield that

$$\begin{aligned} I(x)=\mu _{3}(K^{*})\beta ^{\prime \prime }\left( x\right) h_{1}^{2}/6+u_{p}(h_{1}^{2}), \end{aligned}$$

(37)

in which $\mu _{3}\left( K^{*}\right) =\int v^{3}K^{*}\left( v\right) \mathrm{d}v$. Likewise,

$$\begin{aligned} II(x) =\,&n^{-1}h_{1}^{-1}f^{-1}\left( x\right) \sum \nolimits _{i=1}^{n}K_{h_{1}}^{*}(X_{i}-x)\sigma \left( X_{i}\right) \varepsilon _{i}\left\{ 1+u_{p}\left( \log ^{-1}n\right) \right\} \nonumber \\ =\,&n^{-1}h_{1}^{-1}f^{-1}\left( x\right) \sum \nolimits _{i=1}^{n}K_{h_{1}}^{*}(X_{i}-x)\sigma \left( X_{i}\right) \varepsilon _{i} \nonumber \\&+\,u_{p}\left( n^{-1/2}h_{1}^{-3/2}\log ^{-1/2}n\right) . \end{aligned}$$

(38)

Putting together (33), (37) and (38) completes the proof of the lemma.

Now from Lemma 3, one can rewrite (32) as

$$\begin{aligned} \tilde{\rho }_{\mathrm {LQ}}\left( x\right) -\rho (x) =\,&\sigma _{1}\left\{ 1-\rho ^{2}\left( x\right) \right\} ^{3/2}\sigma ^{-1}\left( x\right) \mu _{3}(K^{*})\beta ^{\prime \prime }\left( x\right) h_{1}^{2}/6 \nonumber \\&+\,\sigma _{1}\left\{ 1-\rho ^{2}\left( x\right) \right\} ^{3/2}f^{-1/2}\left( x\right) n^{-1/2}h_{1}^{-3/2}Y(x) \nonumber \\&+\,u_{p}\left( n^{-1/2}h_{1}^{-3/2}\log ^{-1/2}n+h_{1}^{2}\right) , \end{aligned}$$

(39)

in which the process

$$\begin{aligned} Y(x)=h_{1}^{1/2}\sigma ^{-1}(x)f^{-1/2}\left( x\right) n^{-1/2}\sum \nolimits _{i=1}^{n}K_{h_{1}}^{*}(X_{i}-x)\sigma \left( X_{i}\right) \varepsilon _{i},x\in \mathcal {I}_{n}. \end{aligned}$$

(40)

Define next four stochastic processes, which approximate each other in probability uniformly over $\mathcal {I} _{n}$ or have the exact same distributions over $\mathcal {I}_{n}$. More precisely, with $D_{n}$ defined in Lemma 2, and $B_{n}$ in Lemma 1, let

$$\begin{aligned} Y_{0}(x)= & {} h_{1}^{1/2}\sigma ^{-1}(x)f^{-1/2}\left( x\right) \int _{\mathbb {R} }K_{h_{1}}^{*}(u-x)\varepsilon I_{\left\{ \left| \varepsilon \right| \le D_{n}\right\} }\mathrm{d}B_{n}\left\{ M(u,\varepsilon )\right\} , \end{aligned}$$

(41)

$$\begin{aligned} Y_{1}(x)= & {} h_{1}^{1/2}\sigma ^{-1}(x)f^{-1/2}\left( x\right) \int _{\mathbb {R} }K_{h_{1}}^{*}(u-x)\varepsilon I_{\left\{ \left| \varepsilon \right| \le D_{n}\right\} }\mathrm{d}W_{n}\left\{ M(u,\varepsilon )\right\} , \end{aligned}$$

(42)

$$\begin{aligned} Y_{2}(x)= & {} h_{1}^{1/2}\sigma ^{-1}(x)f^{-1/2}\left( x\right) \int _{\mathbb {R} }K_{h_{1}}^{*}(u-x)f^{1/2}(u)\sigma (u)s_{n}\left( u\right) \mathrm{d}W_{n}(u), \end{aligned}$$

(43)

where

$$\begin{aligned} s_{n}^{2}(u)=\int _{\mathbb {R}}\varepsilon ^{2}I_{\left\{ \left| \varepsilon \right| \le D_{n}\right\} }\mathrm{d}F\left( \varepsilon \left| u\right. \right) , \end{aligned}$$

and satisfies that

$$\begin{aligned}&\sup _{u\in \mathcal {I}_{n}}\left| s_{n}^{2}\left( u\right) -1\right| =\sup _{u\in \mathcal {I}_{n}}\int _{\mathbb {R}}\varepsilon ^{2}I_{\left\{ \left| \varepsilon \right| >D_{n}\right\} }\mathrm{d}F\left( \varepsilon \left| u\right. \right) \le M_{\eta _{1}}D_{n}^{-\eta _{1}}, \end{aligned}$$

(44)

$$\begin{aligned}&Y_{3}(x)=h_{1}^{1/2}\int _{\mathbb {R}}K_{h_{1}}^{*}\left( u-x\right) \mathrm{d}W_{n}\left( u\right) . \end{aligned}$$

(45)

Lemma 4

Under Assumptions (A2)–(A5), as $n\rightarrow \infty ,$

$$\begin{aligned} \sup _{x\in \mathcal {I}_{n}}\left| Y\left( x\right) -Y^{D}\left( x\right) \right| =\mathcal {O}_{p}\left( n^{1/2}h_{1}^{1/2}D_{n}^{-1-\eta _{1}}\right) , \end{aligned}$$

where, for $x\in \mathcal {I}_{n},$

$$\begin{aligned} Y^{D}\left( x\right)= & {} h_{1}^{1/2}\sigma ^{-1}(x)f^{-1/2}\left( x\right) n^{-1/2} \nonumber \\&\times \sum \nolimits _{i=1}^{n}K_{h_{1}}^{*}(X_{i}-x)\sigma \left( X_{i}\right) \varepsilon _{i}I_{\left\{ \left| \varepsilon \right| \le D_{n}\right\} }. \end{aligned}$$

(46)

Proof

Using notations from Lemma 1, the processes Y(x) defined in (40) and $Y^{D}(x)$ can be written as

$$\begin{aligned} Y(x)= & {} h_{1}^{1/2}\sigma ^{-1}(x)f^{-1/2}\left( x\right) \iint K_{h_{1}}^{*}(u-x)\sigma \left( u\right) \varepsilon \mathrm{d}Z_{n}\left( u,\varepsilon \right) , \\ Y^{D}(x)= & {} h_{1}^{1/2}\sigma ^{-1}(x)f^{-1/2}\left( x\right) \iint K_{h_{1}}^{*}(u-x)\sigma \left( u\right) \varepsilon I_{\left\{ \left| \varepsilon \right| \le D_{n}\right\} }\mathrm{d}Z_{n}\left( u,\varepsilon \right) . \end{aligned}$$

The tail part $Y\left( x\right) -Y^{D}\left( x\right)$ is bounded uniformly over $\mathcal {I}_{n}$ by

$$\begin{aligned}&\sup _{x\in \mathcal {I}_{n}}h_{1}^{1/2}\sigma ^{-1}(x)f^{-1/2}\left( x\right) \left| \iint K_{h_{1}}^{*}(u-x)\sigma \left( u\right) \varepsilon I_{\left\{ \left| \varepsilon \right|>D_{n}\right\} }\mathrm{d}Z_{n}\left( u,\varepsilon \right) \right| \nonumber \\&\quad \le \sup _{x\in \mathcal {I}_{n}}h_{1}^{1/2}\sigma ^{-1}(x)f^{-1/2}\left( x\right) n^{-1/2} \nonumber \\&\quad \quad \times \, \left| \sum \nolimits _{i=1}^{n}K_{h_{1}}^{*}\left( X_{i}-x\right) \sigma \left( X_{i}\right) \varepsilon _{i}I_{\left\{ \left| \varepsilon _{i}\right| >D_{n}\right\} }\right| \end{aligned}$$

(47)

$$\begin{aligned}&\quad \quad +\sup _{x\in \mathcal {I}_{n}}n^{1/2}h_{1}^{1/2}\sigma ^{-1}(x)f^{-1/2}\left( x\right) \nonumber \\&\quad \quad \times \,\left| \iint K_{h_{1}}^{*}(u-x)\sigma \left( u\right) \varepsilon I_{\left\{ \left| \varepsilon \right| >D_{n}\right\} }\mathrm{d}F\left( u,\varepsilon \right) \right| . \end{aligned}$$

(48)

By (31) in Lemma 2 and Borel–Cantelli Lemma, the first term in Eq. (47) is $\mathcal {O}_{a.s.}\left( n^{-a}\right)$ for any $a>0$, for instance $a=100$, and the second term in Eq. (48) is bounded by

$$\begin{aligned}&\sup _{x\in \mathcal {I}_{n}}n^{1/2}h_{1}^{1/2}f^{-1/2}\left( x\right) \sigma ^{-1}(x) \\&\quad \quad \times \, \int \left| K_{h_{1}}^{*}(u-x)\right| \sigma \left( u\right) f\left( u\right) \left[ \int \left| \varepsilon \right| I_{\left\{ \left| \varepsilon \right| > D_{n}\right\} }\mathrm{d}F\left( \varepsilon \left| u\right. \right) \right] \mathrm{d}u \\&\quad \le \sup _{x\in \mathcal {I}_{n}}n^{1/2}h_{1}^{1/2}f^{-1/2}\left( x\right) \sigma ^{-1}(x)M_{\eta _{1}}D_{n}^{-\left( 1+\eta _{1}\right) }\int \left| K_{h_{1}}^{*}(u-x)\right| \sigma \left( u\right) f\left( u\right) \mathrm{d}u \\&\quad \le Cn^{1/2}h_{1}^{1/2}D_{n}^{-1-\eta _{1}}=\mathcal {O}\left( n^{1/2}h_{1}^{1/2}D_{n}^{-1-\eta _{1}}\right) . \end{aligned}$$

Lemma 5

Under Assumptions (A2)–(A5), as $n\rightarrow \infty ,$

$$\begin{aligned} \sup _{x\in \mathcal {I}_{n}}\left| Y^{D}\left( x\right) -Y_{0}\left( x\right) \right| =\mathcal {O}_{p}\left( n^{-1/2}h_{1}^{-1/2}D_{n}\log ^{2}n\right) . \end{aligned}$$

Proof

First, $\left| Y^{D}\left( x\right) -Y_{0}\left( x\right) \right|$ can be written as

$$\begin{aligned} h_{1}^{1/2}\sigma ^{-1}(x)f^{-1/2}\left( x\right) \iint K_{h_{1}}^{*}(u-x)\sigma \left( u\right) \varepsilon I_{\left\{ \left| \varepsilon \right| \le D_{n}\right\} }d\left[ Z_{n}\left( u,\varepsilon \right) -B_{n}\left\{ M(u,\varepsilon )\right\} \right] , \end{aligned}$$

which becomes the following via integration by parts

$$\begin{aligned}&h_{1}^{1/2}\sigma ^{-1}(x)f^{-1/2}\left( x\right) \\&\quad \times \iint \sigma \left( u\right) \left[ Z_{n}\left( u,\varepsilon \right) -B_{n}\left\{ M\left( u,\varepsilon \right) \right\} \right] d\left\{ \varepsilon I_{\left\{ \left| \varepsilon \right| \le D_{n}\right\} }\right\} d\left\{ K_{h_{1}}^{*}\left( u-x\right) \right\} . \end{aligned}$$

Next, from the strong approximation result in Eq. (30) and the first condition in Lemma 2, $\sup \nolimits _{x\in \mathcal {I}_{n}}\left| Y^{D}\left( x\right) -Y_{0}\left( x\right) \right|$ is bounded by

$$\begin{aligned} \mathcal {O}_{a.s.}\left( h_{1}^{1/2}h_{1}^{-2}n^{-1/2}h_{1}D_{n}\log ^{2}n\right) =\mathcal {O}_{a.s.}\left( n^{-1/2}h_{1}^{-1/2}D_{n}\log ^{2}n\right) , \end{aligned}$$

thus completing the proof of the lemma.

Lemma 6

Under Assumptions (A2)–(A5), as $n\rightarrow \infty ,$

$$\begin{aligned} \sup _{x\in \mathcal {I}_{n}}\left| Y_{0}\left( x\right) -Y_{1}\left( x\right) \right| =\mathcal {O}_{p}\left( h_{1}^{1/2}\right) . \end{aligned}$$

Proof

Based on Rosenblatt transformation $M(x,\varepsilon )$ defined in Eq. (29) and according to Lemma , the term $\left| Y_{0}\left( x\right) -Y_{1}\left( x\right) \right|$ is bounded by

$$\begin{aligned}&\sup _{x\in \mathcal {I}_{n}}h_{1}^{1/2}\sigma ^{-1}(x)f^{-1/2}\left( x\right) \\&\quad \quad \times \,\left| \iint K_{h_{1}}^{*}\left( u-x\right) \sigma \left( u\right) \left| \varepsilon \right| I_{\left\{ \left| \varepsilon \right| \le D_{n}\right\} }\mathrm{d}M(u,\varepsilon )W_{n}\left( 1,1\right) \right| \\&\quad \le \sup _{x\in \mathcal {I}_{n}}h_{1}^{1/2}\sigma ^{-1}(x)f^{-1/2}\left( x\right) \left| W_{n}\left( 1,1\right) \right| \\&\quad \quad \times \int \left| K_{h_{1}}^{*}\left( u-x\right) \right| \sigma \left( u\right) f\left( u\right) \mathrm{d}u\left\{ \int \left| \varepsilon \right| I_{\left\{ \left| \varepsilon \right| \le D_{n}\right\} }\mathrm{d}F\left( \varepsilon \left| u\right. \right) \right\} = \mathcal {O}_{p}\left( h_{1}^{1/2}\right) . \end{aligned}$$

The next lemma expresses the distribution of $Y_{1}\left( x\right)$ in terms of one-dimensional Brownian motion.

Lemma 7

The process $Y_{1}\left( x\right)$ has the same distribution as $Y_{2}\left( x\right)$ over $x\in \mathcal {I}_{n}.$

Proof

By definitions, $Y_{1}\left( x\right)$ defined in (42) and $Y_{2}\left( x\right)$ in (43) are Gaussian processes with zero mean and unit variance. They have the same covariance functions as

$$\begin{aligned} \mathrm {cov}\left\{ Y_{1}\left( x\right) ,Y_{1}\left( x^{\prime }\right) \right\} =\,&h_{1}^{1/2}\sigma ^{-1}(x)f^{-1/2}\left( x\right) h_{1}^{1/2}\sigma ^{-1}(x^{\prime })f^{-1/2}\left( x^{\prime }\right) \\&\times \int K_{h_{1}}^{*}(u-x)K_{h_{1}}^{*}(u-x^{\prime })f(u)\sigma ^{2}(u)s_{n}^{2}\left( u\right) \mathrm{d}u \\ =\,&\mathrm {cov}\left\{ Y_{2}\left( x\right) ,Y_{2}\left( x^{\prime }\right) \right\} . \end{aligned}$$

Hence, according to Itô’s Isometry Theorem, they have the same distribution.

Lemma 8

Under Assumptions (A2)–(A5), as $n\rightarrow \infty ,$

$$\begin{aligned} \sup _{x\in \mathcal {I}_{n}}\left| Y_{2}\left( x\right) -Y_{3}\left( x\right) \right| =\mathcal {O}_{p}\left( h_{1}^{1/2}+h_{1}^{-1/2}D_{n}^{-\eta _{1}}\right) . \end{aligned}$$

Proof

By the aforementioned condition in Lemma 2 and Eq. (44), $\sup \nolimits _{x\in \mathcal {I}_{n}}\left| Y_{2}\left( x\right) \right. \left. -Y_{3}\left( x\right) \right|$ is almost surely bounded by

$$\begin{aligned}&\sup _{x\in \mathcal {I}_{n}}\left| W_{n}\left( u\right) \right| h_{1}^{1/2}\left| \int d\left[ K_{h_{1}}^{*}\left( u-x\right) \left[ \left\{ \frac{f\left( u\right) }{f\left( x\right) }\right\} ^{1/2}\left\{ \frac{\sigma \left( u\right) }{\sigma \left( x\right) }\right\} s_{n}\left( u\right) -1\right] \right] \right| \\&\quad \le \sup _{x\in \mathcal {I}_{n}}\left| W_{n}\left( u\right) \right| h_{1}^{1/2}h_{1}^{-1} \\&\quad \quad \times \int h_{1}^{-1}\left| K^{*\prime }\left( \frac{u-x}{h_{1}} \right) \right| \left[ \left\{ \frac{f\left( u\right) }{f\left( x\right) }\right\} ^{1/2}\left\{ \frac{\sigma \left( u\right) }{\sigma \left( x\right) }\right\} s_{n}\left( u\right) -1\right] \\&\quad \quad +\,\left| K^{*}\left( \frac{u-x}{h_{1}}\right) \right| \left[ \left\{ \frac{f\left( u\right) }{f\left( x\right) }\right\} ^{1/2}\left\{ \frac{\sigma \left( u\right) }{\sigma \left( x\right) }\right\} s_{n}\left( u\right) -1\right] ^{^{\prime }}\mathrm{d}u \\&\quad =\mathcal {O}_{p}\left( h_{1}^{-1/2}\right) \left\{ III\left( x\right) +IV\left( x\right) \right\} , \end{aligned}$$

where the term $III\left( x\right)$ is bounded by

$$\begin{aligned}&\sup _{x\in \mathcal {I}_{n}}Ch_{1}^{-1}\left\| K^{*}\right\| _{\infty }f^{-1/2}\left( x\right) \sigma ^{-1}\left( x\right) h_{1} \\&\qquad \times \, \left| \left[ h_{1}/2\left\{ f^{\prime }\left( x\right) \right\} ^{1/2}\sigma \left( u\right) s_{n}\left( u\right) +f^{1/2}\left( x\right) \left\{ \sigma \left( u\right) s_{n}\left( u\right) -\sigma \left( x\right) \right\} \right] \right| \\&\quad \le C\left\| K^{*}\right\| _{\infty }C_{f}^{-1/2}C_{\sigma }^{-1}\left\{ h_{1}/2+\left\| s_{n}^{2}-1\right\| _{\infty }\right\} \\&\quad \le C\left( h_{1}+D_{n}^{-\eta _{1}}\right) , \end{aligned}$$

and the term $IV\left( x\right)$ is bounded by

$$\begin{aligned}&Ch_{1}ch_{1}2^{-1}f^{\prime }\left( u\right) f^{-1/2}\left( u\right) f^{-1/2}\left( x\right) \sigma \left( u\right) \sigma ^{-1}\left( x\right) s_{n}\left( u\right) \\&\qquad +\,Ch_{1}f^{1/2}\left( u\right) f^{-1/2}\left( x\right) \left\{ \sigma ^{\prime }\left( u\right) \sigma ^{-1}\left( x\right) s_{n}\left( u\right) +\sigma \left( u\right) s_{n}^{\prime }\left( u\right) /\sigma \left( x\right) \right\} \\&\quad \le Ch_{1}\left( 2^{-1}\left\| f^{\prime }\right\| _{\infty }C_{f}\left\| s_{n}\right\| _{\infty }+\left\| \sigma ^{\prime }\right\| _{\infty }C_{\sigma }^{-1}\left\| s_{n}\right\| _{\infty }+\left\| s_{n}^{\prime }\right\| _{\infty }\right) \\&\quad \le Ch_{1}. \end{aligned}$$

Putting together the above, one obtains that

$$\begin{aligned} \sup \nolimits _{x\in \mathcal {I}_{n}}\left| Y_{2}\left( x\right) -Y_{3}\left( x\right) \right|= & {} \mathcal {O}_{p}\left( h_{1}^{-1/2}\right) \left\{ III\left( x\right) +IV\left( x\right) \right\} \\= & {} \mathcal {O}_{p}\left( h_{1}^{1/2}+h_{1}^{-1/2}D_{n}^{-\eta _{1}}\right) + \mathcal {O}_{p}\left( h_{1}^{1/2}\right) , \end{aligned}$$

completing the proof of this lemma.

Proof of Proposition 1

The absolute maximum of $\left\{ Y_{3}\left( x\right) ,x\in \mathcal {I}_{n}\right\}$ is the same as that of

$$\begin{aligned}&\left\{ h_{1}^{-1/2}\int K^{*}\left( \frac{u}{h_{1}}-x\right) \mathrm{d}W_{n}\left( u\right) ,x\in \left[ ah_{1}^{-1}+1,bh_{1}^{-1}-1\right] \right\} \nonumber \\&\quad =\left\{ \int K^{*}\left( v-x\right) \mathrm{d}W_{n}\left( v\right) \mathrm {, }x\in \left[ ah_{1}^{-1}+1,bh_{1}^{-1}-1\right] \right\} . \end{aligned}$$

(49)

For process $\xi \left( x\right) =\int K^{*}(v-x)\mathrm{d}W_{n}\left( v\right)$, $x\in \left[ ah_{1}^{-1}+1, bh_{1}^{-1}-1\right]$, the correlation function is

$$\begin{aligned} r\left( x-y\right) =\frac{\mathsf {E}\left\{ \xi \left( x\right) \xi \left( y\right) \right\} }{\mathrm {var}^{1/2}\left\{ \xi \left( x\right) \right\} \mathrm {var} ^{1/2}\left\{ \xi \left( y\right) \right\} }, \end{aligned}$$

which implies that

$$\begin{aligned} r\left( t\right) =\frac{\int K^{*}\left( v\right) K^{*}\left( v-t\right) \mathrm{d}v}{\int K^{*}\left( v\right) ^{2}\mathrm{d}v}. \end{aligned}$$

Define next a Gaussian process $\varsigma \left( t\right) ,0\le t\le T=T_{n}=\left( b-a\right) /h_{1}-2,$

$$\begin{aligned} \varsigma \left( t\right) =\xi \left( t+ah_{1}^{-1}+1\right) \left\{ \int K^{*}\left( v\right) ^{2}\mathrm{d}v\right\} ^{-1/2}, \end{aligned}$$

which is stationary with mean zero and variance one, and covariance function

$$\begin{aligned} r\left( t\right) =\mathsf {E}\varsigma \left( s\right) \varsigma \left( t+s\right) =1-Ct^{2}+o\left( \left| t\right| ^{2}\right) \mathrm { as } t\rightarrow 0, \end{aligned}$$

with $C=C_{K^{*\prime }}/2C_{K^{*}}$. Then applying Theorems 11.1.5 and 12.3.5 of Leadbetter et al. (1983), one has for $h_{1}\rightarrow 0$ or $T\rightarrow \infty$,

$$\begin{aligned} \mathbb {P}\left[ a_{T}\left\{ \sup \nolimits _{t\in \left[ 0,T\right] }\left| \varsigma \left( t\right) \right| -b_{T}\right\} \le z \right] \rightarrow e^{-2e^{-z}},\quad \forall z\in \mathbb {R}, \end{aligned}$$

where $a_{T}=\left( 2\log T\right) ^{1/2}$ and $b_{T}=$ $a_{T}+a_{T}^{-1} \left\{ \sqrt{C\left( K^{*}\right) }/2\pi \right\}$. Note that for $a_{h_{1}},b_{h_{1}}$ defined in (21), as $n\rightarrow \infty$,

$$\begin{aligned} a_{h_{1}}a_{T}^{-1}\rightarrow 1,a_{T}\left( b_{T}-b_{h_{1}}\right) = \mathcal {O}\left( \log ^{1/2}n\times h_{1}\log ^{-1/2}n\right) \rightarrow 0. \end{aligned}$$

Hence, applying Slutsky’s Theorem twice, one obtains that

$$\begin{aligned} a_{h_{1}}\left\{ \sup \nolimits _{t\in \left[ 0,T\right] }\left| \varsigma \left( t\right) \right| -b_{h_{1}}\right\}= & {} a_{h_{1}}a_{T}^{-1}\left[ a_{T}\left\{ \sup \nolimits _{t\in \left[ 0,T\right] }\left| \varsigma \left( t\right) \right| -b_{T}\right\} \right] \\&+\,a_{h_{1}}\left( b_{T}-b_{h_{1}}\right) \end{aligned}$$

converges in distribution to the same limit as $a_{T}\left\{ \sup \nolimits _{t\in \left[ 0,T\right] }\left| \varsigma \left( t\right) \right| -b_{T}\right\}$. Thus,

$$\begin{aligned} \mathbb {P}\left( a_{h_{1}}\left[ \frac{\sup \nolimits _{x\in \mathcal {I} _{n}}\left| Y_{3}\left( x\right) \right| }{\left\{ \int K^{*}\left( v\right) ^{2}\mathrm{d}v\right\} ^{1/2}}-b_{h_{1}}\right] \le z\right) \rightarrow e^{-2e^{-z}},\quad \forall z\in \mathbb {R}. \end{aligned}$$

Next applying Lemma 8 and Slutsky’s Theorem, $\forall z\in \mathbb {R},$

$$\begin{aligned} \mathbb {P}\left( a_{h_{1}}\left[ \frac{\sup \nolimits _{x\in \mathcal {I} _{n}}\left| Y_{2}\left( x\right) \right| }{\left\{ \int K^{*}\left( v\right) ^{2}\mathrm{d}v\right\} ^{1/2}}-b_{h_{1}}\right] \le z\right) \rightarrow e^{-2e^{-z}}. \end{aligned}$$

(50)

Furthermore, applying Lemma 7 and Slutsky’s Theorem, the limiting distribution (50) is the same as

$$\begin{aligned} \mathbb {P}\left( a_{h_{1}}\left[ \frac{\sup \nolimits _{x\in \mathcal {I} _{n}}\left| Y_{1}\left( x\right) \right| }{\left\{ \int K^{*}\left( v\right) ^{2}\mathrm{d}v\right\} ^{1/2}}-b_{h_{1}}\right] \le z\right) \rightarrow e^{-2e^{-z}}. \end{aligned}$$

Furthermore, applying Lemmas 1 to 6 and Slutsky’s Theorem, one obtains

$$\begin{aligned} \mathbb {P}\left( a_{h_{1}}\left[ \frac{\sup \nolimits _{x\in \mathcal {I} _{n}}\left| Y\left( x\right) \right| }{\left\{ \int K^{*}\left( v\right) ^{2}\mathrm{d}v\right\} ^{1/2}}-b_{h_{1}}\right] \le z\right) \rightarrow e^{-2e^{-z}}. \end{aligned}$$

(51)

By taking $1-\alpha =e^{-2e^{-z}}$ for $\alpha \in \left( 0,1\right)$, the above (51) implies that

$$\begin{aligned} \lim _{n\rightarrow \infty }\mathbb {P}\left\{ \rho \left( x\right) \in \tilde{ \rho }_{\mathrm {\mathrm {LQ}}}(x)\pm a_{h_{1}}V_{n}(x)Q_{n}\left( \alpha \right) ,x\in \mathcal {I}_{n}\right\} =1-\alpha . \end{aligned}$$

Thus, an infeasible SCB for $\rho \left( x\right)$ over $\mathcal {I}_{n}$ is

$$\begin{aligned} \tilde{\rho }_{\mathrm {\mathrm {LQ}}}(x)\pm a_{h_{1}}V_{n}(x)Q_{n}\left( \alpha \right) , \end{aligned}$$

which establishes Proposition 1.

Proof of Theorem 1

Applying Taylor expansion to $\hat{\rho }_{\mathrm {\mathrm {LQ}}}(x)-\tilde{\rho }_{\mathrm {LQ} }\left( x\right)$, its asymptotic order is the lower of $\hat{\sigma } _{1}^{2}-\sigma _{1}^{2}$ and $\hat{\sigma }_{\mathrm {SK}}^{2}(x)-\sigma ^{2}\left( x\right)$. While $\hat{\sigma }_{1}^{2}-\sigma _{1}^{2}=\mathcal {O }_{p}\left( n^{-1/2}\right) ,\sup \nolimits _{x\in \left[ a+h_{2},b-h_{2} \right] }\left| \hat{\sigma }_{\mathrm {SK}}^{2}(x)-\sigma ^{2}\left( x\right) \right|$ is of order $\mathcal {O}_{p}\left( n^{-1/2}h_{2}^{-1/2}\log ^{1/2}n\right)$ according to Cai and Yang (2015), and of order $o_{p}\left( n^{-1/2}h_{1}^{-3/2}\log ^{-1/2}n\right)$ by applying (16). As (17) entails that $\mathcal {I}_{n}\subset \left[ a+h_{2},b-h_{2}\right]$ for large enough n, one has

$$\begin{aligned} \sup \limits _{x\in \mathcal {I}_{n}}\left| \hat{\sigma }_{\mathrm {SK} }^{2}(x)-\sigma ^{2}\left( x\right) \right| =o_{p}\left( n^{-1/2}h_{1}^{-3/2}\log ^{-1/2}n\right) , \end{aligned}$$

(52)

and thus $\sup \nolimits _{x\in \mathcal {I}_{n}}\left| \hat{\rho }_{\mathrm { \mathrm {LQ}}}(x)-\tilde{\rho }_{\mathrm {LQ}}\left( x\right) \right| =o_{p}\left( n^{-1/2}h_{1}^{-3/2}\log ^{-1/2}n\right)$. Hence, the proof of the theorem is complete.

Proof of Theorem 2

Proposition 1, Theorem 1, and repeated applications of Slutsky’s Theorem entail that

$$\begin{aligned} \lim _{n\rightarrow \infty }\mathbb {P}\left\{ \rho \left( x\right) \in \hat{ \rho }_{\mathrm {\mathrm {LQ}}}\left( x\right) \pm a_{h_{1}}\hat{V} _{n}(x)Q_{n}\left( \alpha \right) ,x\in \mathcal {I}_{n}\right\} =1-\alpha , \end{aligned}$$

which yields the oracle SCB for $\rho \left( x\right)$ over $\mathcal {I} _{n}$ in Theorem 2.