Nearest neighbors estimation for long memory functional data

Abstract

In this paper, we consider the asymptotic properties of the nearest neighbors estimation for long memory functional data. Under some regularity assumptions, we investigate the asymptotic normality and the uniform consistency of the nearest neighbors estimators for the nonparametric regression models when the explanatory variable and the errors are of long memory and the explanatory variable takes values in some abstract functional space. The finite sample performance of the proposed estimator is discussed through simulation studies.

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Correspondence to Lihong Wang.

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This work was supported by National Natural Science Foundation of China (NSFC) Grants 11671194 and 11501287.

Appendix

Appendix

Proof of Lemma 1

It suffices to show that, as \(n\rightarrow \infty \),

$$\begin{aligned} \text {E}[g_n(x,H_n)]\longrightarrow 0 \end{aligned}$$
(10)

and

$$\begin{aligned} \text {Var}[g_n(x,H_n)]\longrightarrow 0. \end{aligned}$$
(11)

Let \(\xi _{n1}=\min (H_n, h_n)\) and \(\xi _{n2}=\max (H_n, h_n)\). Then, by (8), \(\xi _{n1}=h_n(1+ o(n^{-\rho }))\), \(\xi _{n2}=h_n(1+ o(n^{-\rho }))\) and \(\xi _{n2}-\xi _{n1}=o(h_nn^{-\rho })\) a.s.

Since E\(\varepsilon _0=0\) and \(k_n=nh_nf_x(0)\), by Assumption (A4),

$$\begin{aligned} \text {E}[g_n(x, H_n)]= & {} \text {E}\big [n^{1/2-\beta }k_n^{-1}\sum _{i=1}^n(K(d(x,X_i)/H_n)-K(d(x,X_i)/h_n))(Y_i-r(x))\big ]\nonumber \\= & {} n^{3/2-\beta }k_n^{-1} \text {E}[(K(d(x,X_1)/H_n)-K(d(x,X_1)/h_n))(r(X_i)-r(x))]\nonumber \\\le & {} Cn^{3/2-\beta }k_n^{-1}\text {E}\Big [\int _{\xi _{n1}}^{\xi _{n2}}u dF_x(u)\Big ]\nonumber \\= & {} Cn^{3/2-\beta }k_n^{-1} \text {E}[(\xi _{n2}-\xi _{n1})\zeta _n f_x(\zeta _n)]\nonumber \\= & {} C n^{3/2-\beta }k_n^{-1}h_n o(n^{-\rho }) \text {E}[\zeta _n f_x(\zeta _n)]\nonumber \\= & {} o(n^{\alpha -\beta -1/2-\rho })\text {E}[f_x(\zeta _n)]/f^2_x(0), \end{aligned}$$
(12)

where \(\xi _{n1}<\zeta _n<\xi _{n2}\). Hence (12), together with Assumption (A2) and the fact that \(\zeta _n=h_n(1+o(1))\) a.s. implies (10).

Let \(Z_{i1}=(K(d(x,X_i)/H_n)-K(d(x,X_i)/h_n))(r(X_i)-r(x))\) and \(Z_{i2}=(K(d(x,X_i)/H_n)-K(d(x,X_i)/h_n))\varepsilon _i\). To prove (11), it is enough to show

$$\begin{aligned} \text {Var}\Big (n^{1/2-\beta -\alpha }\sum _{i=1}^n Z_{i1}\Big )\longrightarrow 0, \quad \text {and} \quad \text {Var}\Big (n^{1/2-\beta -\alpha }\sum _{i=1}^n Z_{i2}\Big )\longrightarrow 0. \end{aligned}$$

Note that

$$\begin{aligned} \text {Var}\Big (n^{1/2-\beta -\alpha }\sum _{i=1}^nZ_{i1}\Big )= & {} n^{2-2\beta -2\alpha }\text {Var}(Z_{01})+n^{1-2\beta -2\alpha }\sum _{i\ne j}\text {Cov}(Z_{i1},Z_{j1}). \end{aligned}$$

For the first term of the above variance, similarly to (12), we obtain

$$\begin{aligned} \text {Var}(Z_{01})\le & {} \text {E}\big [(K(d(x,X_0)/H_n)-K(d(x,X_0)/h_n))(r(X_0)-r(x))\big ]^2\\\le & {} C\text {E}\Big [\int _{\xi _{n1}}^{\xi _{n2}}u^2f_x(u)du\Big ]= C\text {E} [(\xi _{n2}-\xi _{n1})\zeta _n^2 f_x(\zeta _n)]\nonumber \\= & {} C h_n^3 o(n^{-\rho })\text {E}[f_x(\zeta _n)]. \end{aligned}$$

This means that

$$\begin{aligned} n^{2-2\beta -2\alpha }\text {Var}(Z_{01})=o(n^{\alpha -2\beta -1-\rho })=o(1). \end{aligned}$$

In addition, by mean value theorem, \(F_x(\xi _{n2})-F_x(\xi _{n1})=f_x(\zeta _n) (\xi _{n2}-\xi _{n1})\) for some \(\xi _{n1}< \zeta _n<\xi _{n2}\). Moreover, by Assumption (A5),

$$\begin{aligned}&P(u_1\le d(x, X_i)\le u_2, u_1\le d(x, X_j)\le u_2)\\&\quad =P(d(x, X_i)\le u_2, d(x, X_j)\le u_2)+P(d(x, X_i)\le u_1, d(x, X_j)\le u_1)\\&\qquad -\,P( d(x, X_i)\le u_2, d(x, X_j)\le u_1)-P( d(x, X_i)\le u_1, d(x, X_j)\le u_2)\\&\quad \le F_x^2(u_2)+F_x^2(u_1)-2F_x(u_2)F_x(u_1)+C\gamma _x(i-j)\\&\quad =\big (F_x(u_2)-F_x(u_1)\big )^2+C\gamma _x(i-j), \end{aligned}$$

for any \(u_1\), \(u_2\) close to 0. Thus, by (8) and Assumption (A4),

$$\begin{aligned} \text {Cov}(Z_{i1},Z_{j1})= & {} \text {Cov}\left[ (r(X_i)-r(x))(K(d(x,X_i)/H_n)-K(d(x,X_i)/h_n)),\right. \nonumber \\&\left. (r(X_j)-r(x))(K(d(x,X_j)/H_n)-K(d(x,X_j)/h_n))\right] \nonumber \\\le & {} C\text {E}\Big [d(x, X_i)d(x, X_j)\big |K(d(x,X_i)/H_n)-K(d(x,X_i)/h_n)\big |\nonumber \\&\cdot \,\big |K(d(x,X_j)/H_n)-K(d(x,X_j)/h_n)\big |\Big ]\nonumber \\&+\,C\Big (\text {E} \big [d(x, X_i)\big |K(d(x,X_i)/H_n)-K(d(x,X_i)/h_n)\big |\big ]\Big )^2\nonumber \\\le & {} C\text {E}\big [\xi _{n2}^2P(\xi _{n1}\le d(x, X_i)\le \xi _{n2}, \xi _{n1}\le d(x, X_j)\le \xi _{n2})\big ]\nonumber \\&+\,C\Big (\text {E}\big [\xi _{n2} P(\xi _{n1}\le d(x, X_0)\le \xi _{n2})\big ]\Big )^2\nonumber \\\le & {} C h_n^2(1+o(1))\Big \{\gamma _x(i-j)+\text {E}\big [F_x(\xi _{n2})-F_x(\xi _{n1})\big ]^2\Big \}\nonumber \\= & {} C h_n^2(1+o(1))\Big \{\gamma _x(i-j)+o(h_n^2 n^{-2\rho })\Big \}. \end{aligned}$$

Now using (7) and Assumption (A2), we arrive at, for some large enough N,

$$\begin{aligned} n^{1-2\beta -2\alpha }\sum _{i\ne j}\text {Cov}(Z_{i1},Z_{j1})= & {} Cn^{1-2\beta -2\alpha }h_n^2\Big \{\sum _{i\ne j}\gamma _x(i-j)+o(n^{2\alpha -2\rho })\Big \}\\= & {} O(n^{-2\beta -1})\Big \{n\sum _{k=1}^n\gamma _x(k)+o(n^{2\alpha -2\rho })\Big \}\\= & {} O(n^{-2\beta -1})\Big \{n\big (\sum _{k=1}^N+\sum _{k=N+1}^n\big )\gamma _x(k)+o(n^{2\alpha -2\rho })\Big \}\\\le & {} O(n^{-2\beta -1})\Big \{n\sum _{k=1}^n k^{-\tau _x D}+o(n^{2\alpha -2\rho })\Big \}\\= & {} O(n^{-2\beta -1})\Big \{n^{2-\tau _x D}+o(n^{2\alpha -2\rho })\Big \}=o(1). \end{aligned}$$

These bounds imply the weak convergence of \(n^{1/2-\beta -\alpha }\sum _{i=1}^n Z_{i1}\). To derive the same result for \(n^{1/2-\beta -\alpha }\sum _{i=1}^n Z_{i2}\), note that, by (2),

$$\begin{aligned}&\text {Var}\Big (n^{1/2-\beta -\alpha }\sum _{i=1}^nZ_{i2}\Big )\\&\quad = n^{2-2\beta -2\alpha }\text {Var}(Z_{02})+n^{1-2\beta -2\alpha }\sum _{i\ne j}\text {Cov}(Z_{i2},Z_{j2})\\&\quad = n^{2-2\beta -2\alpha }\text {E}\varepsilon _0^2\text {E}\big (K(d(x,X_0)/H_n)-K(d(x,X_0)/h_n)\big )^2\\&\qquad +\,n^{1-2\beta -2\alpha }\sum _{i\ne j}\gamma _\varepsilon (i-j)\text {E}\big [(K(d(x,X_i)/H_n)-K(d(x,X_i)/h_n))\\&\qquad \cdot \, (K(d(x,X_j)/H_n)-K(d(x,X_j)/h_n))\big ]\\&\quad \le Cn^{2-2\beta -2\alpha }\text {E}\big [P(\xi _{n1}\le d(x, X_0)\le \xi _{n2})\big ]\\&\qquad +\,n^{1-2\beta -2\alpha }\sum _{i\ne j}\gamma _\varepsilon (i-j)\text {E}\big [P(\xi _{n1}\le d(x, X_i)\le \xi _{n2}, \xi _{n1}\le d(x, X_j)\le \xi _{n2})\big ]\\&\quad \le C n^{2-2\beta -2\alpha }\text {E}(\xi _{n2}-\xi _{n1})\\&\qquad +\,Cn^{1-2\beta -2\alpha }\sum _{i\ne j}\gamma _\varepsilon (i-j)\big (\gamma _x(i-j)+\text {E} [F_x(\xi _{n2})-F_x(\xi _{n1})]^2\big )\\&\quad = o(n^{1-2\beta -\alpha -\rho })+O(n^{1-2\beta -2\alpha })\big (O(n^{2\beta +1-\tau _x D})+o(n^{2\beta +2\alpha -1-2\rho })\big )\\&\quad =o(n^{1-2\beta -\alpha -\rho })+O(n^{2-2\alpha -\tau _x D})+o(n^{-2\rho }). \end{aligned}$$

Since \(\rho >0\) and \(\tau _x\ge 1\), Assumption (A2) implies that \(1-2\beta -\alpha -\rho <0\) and \(2-2\alpha -\tau _x D<0\). That is,

$$\begin{aligned} \text {Var}\Big (n^{1/2-\beta -\alpha }\sum _{i=1}^nZ_{i2}\Big )=o(1). \end{aligned}$$

This completes the proof of Lemma 1. \(\square \)

Proof of Theorem 1

By (1) and (6),

$$\begin{aligned}&n^{1/2-\beta }(r_n(x,H_n)-r(x))\nonumber \\&\quad =n^{1/2-\beta }k_n^{-1}\sum _{i=1}^n K(d(x,X_i)/h_n)(Y_i-r(x))+g_n(x, H_n)\nonumber \\&\quad =n^{1/2-\beta }k_n^{-1}\sum _{i=1}^n K(d(x,X_i)/h_n)(r(X_i)-r(x)+\varepsilon _i)+g_n(x, H_n)\nonumber \\&\quad =I_{n1}+I_{n2}+g_n(x, H_n) \end{aligned}$$

where \(I_{n1}=n^{1/2-\beta }k_n^{-1}\sum _{i=1}^n K(d(x,X_i)/h_n)(r(X_i)-r(x))\) and \(I_{n2}=n^{1/2-\beta }\) \(k_n^{-1}\) \(\sum _{i=1}^n K(d(x,X_i)/h_n)\varepsilon _i\).

By Lemma 1, it suffices to show that

$$\begin{aligned} I_{n1}{\mathop {\longrightarrow }\limits ^\mathcal{P}} 0, \quad \text {and}\quad I_{n2}{\mathop {\longrightarrow }\limits ^\mathcal{D}} c_0 Z. \end{aligned}$$
(13)

Along the similar lines of the Proof of Lemma 1, we obtain

$$\begin{aligned} \text {E}I_{n1}= & {} n^{1/2-\beta }k_n^{-1}n \text {E}[K(d(x,X_0)/h_n)(r(X_0)-r(x))]\nonumber \\\le & {} Cn^{3/2-\beta -\alpha }\int _0^{h_n} u dF_x(u)=Cn^{3/2-\beta -\alpha } h_n\zeta _nf_x(\zeta _n)\nonumber \\= & {} O(n^{\alpha -\beta -1/2})=o(1), \end{aligned}$$

where \(0<\zeta _n<h_n\). Moreover, again by Assumption (A2),

$$\begin{aligned} \text {Var}(I_{n1})\le & {} n^{2-2\beta -2\alpha }\text {E}\big (K(d(x,X_0)/h_n)(r(X_0)-r(x))\big )^2+n^{1-2\beta -2\alpha }\sum _{i\ne j}\nonumber \\&\text {Cov}\left[ (r(X_i)-r(x))K(d(x,X_i)/h_n), (r(X_j)-r(x))K(d(x,X_j)/h_n)\right] \nonumber \\= & {} n^{2-2\beta -2\alpha }O(h_n^3)+n^{1-2\beta -2\alpha }\Big \{O(h_n^2)\sum _{i\ne j}\gamma _x(i-j)+O(n^2h_n^4)\Big \}\nonumber \\= & {} O(n^{\alpha -2\beta -1})+n^{1-2\beta -2\alpha }\big (O(n^{2\alpha -\tau _x D})+O(n^{4\alpha -2})\big )\nonumber \\= & {} O(n^{\alpha -2\beta -1})+O(n^{1-2\beta -\tau _x D})+O(n^{2\alpha -2\beta -1})=o(1). \end{aligned}$$
(14)

Then we arrive at \(I_{n1}{\mathop {\longrightarrow }\limits ^\mathcal{P}} 0\). For \(I_{n2}\), note that

$$\begin{aligned} I_{n2}= & {} n^{1/2-\beta }k_n^{-1}\sum _{i=1}^n \big (K(d(x,X_i)/h_n)-\text {E}(K(d(x,X_i)/h_n))\big )\varepsilon _i\nonumber \\&+\,n^{1/2-\beta }k_n^{-1}\sum _{i=1}^n \text {E}(K(d(x,X_i)/h_n))\varepsilon _i. \end{aligned}$$
(15)

Since

$$\begin{aligned} \text {E}(K(d(x,X_i)/h_n))=\int _0^{h_n} dF_x(u)=F_x(h_n)=h_nf_x(\zeta _n)=h_n(f_x(0)+o(1)), \end{aligned}$$

where \(0<\zeta _n<h_n\), by (9), we have

$$\begin{aligned} n^{1/2-\beta }k_n^{-1}\sum _{i=1}^n \text {E}(K(d(x,X_i)/h_n))\varepsilon _i {\mathop {\longrightarrow }\limits ^\mathcal{D}} c_0 Z. \end{aligned}$$

Therefore, to complete the proof, it suffices to show the first term of (15) tends to 0 in probability. In a similar way as in (14), the variance for the first term of (15) is bounded by

$$\begin{aligned}&n^{2-2\beta -2\alpha }\text {E}\big (K(d(x,X_0)/h_n)\big )^2\text {E}\varepsilon _0^2\\&\qquad +\,n^{1-2\beta -2\alpha }\sum _{i\ne j}\gamma _{\varepsilon }(i-j) \text {Cov}\left[ K(d(x,X_i)/h_n), K(d(x,X_j)/h_n)\right] \\&\quad \le n^{2-2\beta -2\alpha }O(h_n)+Cn^{1-2\beta -2\alpha }\sum _{i\ne j}\gamma _\varepsilon (i-j)\gamma _x(i-j)\\&\quad =O(n^{1-2\beta -\alpha })+O(n^{2-2\alpha -\tau _x D})=o(1). \end{aligned}$$

This concludes the Proof of Theorem 1. \(\square \)

Proof of Theorem 2

From (6), we obtain

$$\begin{aligned}&r_n(x, H_n)-r(x)\\&\quad =k_n^{-1}\sum _{i=1}^n K(d(x, X_i)/H_n)(r(X_i)-r(x))+k_n^{-1}\sum _{i=1}^n K(d(x, X_i)/H_n)\varepsilon _i\\&\quad :=R_1(x)+R_2(x), \quad \text {say}. \end{aligned}$$

It suffices to show that

$$\begin{aligned} \sup _{x\in S} |R_1(x)|=o_P(1),\quad \text {and}\quad \sup _{x\in S} |R_2(x)|=o_P(1). \end{aligned}$$
(16)

By (8) and Assumption (A4), we have

$$\begin{aligned} \sup _{x\in S} |R_1(x)|\le \sup _{x\in S} \sup _{y\in B(x, H_n)}|r(y)-r(x)|\le CH_n=O(h_n)\quad \text {a.s.} \end{aligned}$$

This is enough to prove the first claim of (16). It just remains to check the second result of (16). Note that we can write

$$\begin{aligned} \sup _{x\in S} |R_2(x)|\le \sup _{x\in S} |R_2(x)-R_2(t_x)|+\sup _{x\in S} |R_2(t_x)|, \end{aligned}$$
(17)

while,

$$\begin{aligned} \sup _{x\in S} |R_2(x)-R_2(t_x)|\le \sup _{x\in S} k_n^{-1}\sum _{i=1}^n \big |K(d(x, X_i)/H_n)-K(d(t_x, X_i)/H_n)\big ||\varepsilon _i|. \end{aligned}$$

By (8) and Assumption (B2), we have

$$\begin{aligned}&\text {E} \Big [\big |K(d(x, X_0)/H_n)-K(d(t_x, X_0)/H_n)\big |\Big ]\nonumber \\&\quad = \text {P}((X_0\in B(x, h_n)\bigcap \bar{B}(t_x, h_n))+\text {P}(X_0\in \bar{B}(x, h_n)\bigcap B(t_x, h_n))\nonumber \\&\quad =O(l_n) \end{aligned}$$

for any \(x\in S\). This result implies that, for any \(x\in S\) and any \(1\le i\le n\),

$$\begin{aligned} \big |K(d(x, X_i)/H_n)-K(d(t_x, X_i)/H_n)\big |=O_P(l_n). \end{aligned}$$

Moreover, for any \(\delta >0\), there exists an \(x^*\in S\) with

$$\begin{aligned}&\sup _{x\in S}\big |K(d(x, X_i)/H_n)-K(d(t_x, X_i)/H_n)\big |\\&\quad < |K(d(x^*, X_i)/H_n)-K(d(t_{x^*}, X_i)/H_n)\big |+\delta . \end{aligned}$$

Let \(\delta \rightarrow 0\), we obtain

$$\begin{aligned} \sup _{x\in S}\big |K(d(x, X_i)/H_n)-K(d(t_x, X_i)/H_n)\big |=O_P(l_n). \end{aligned}$$

This leads directly to

$$\begin{aligned} \sup _{x\in S} |R_2(x)-R_2(t_x)|=O_P(l_nk_n^{-1}n)=O_P(n^{1-\alpha -\xi })=o_P(1). \end{aligned}$$
(18)

Looking at the second term on the RHS of (17), we have, for any \(\varepsilon >0\),

$$\begin{aligned}&\text {P}\big (\sup _{x\in S} |R_2(t_x)|>\varepsilon \big )\\&\quad =\text {P}\big (\max _{k\in \{1,\ldots , {z_n}\} } |R_2(t_k)-\text {E}(R_2(t_k))|>\varepsilon \big )\\&\quad \le z_n\max _{k\in \{1,\ldots , {z_n}\} }\text {P}\big ( |R_2(t_k)-\text {E}(R_2(t_k))|>\varepsilon \big ). \end{aligned}$$

By using the similar lines of the proof of (13), and by Assumptions (A2), (B1) and (B3), one gets directly for any \(\varepsilon >0\),

$$\begin{aligned} \text {P}\big (\sup _{x\in S} |R_2(t_x)|>\varepsilon \big )\le z_n\text {Var}\big ( R_2(t_1)\big )/\varepsilon ^2 =O(l_n^{-1})O(n^{2\beta -1})=o(1). \end{aligned}$$

This leads to

$$\begin{aligned} \sup _{x\in S} |R_2(t_x)|=o_P(1). \end{aligned}$$

This, together with (18), is enough to get

$$\begin{aligned}\sup _{x\in S} |R_2(x)|=o_P(1).\end{aligned}$$

This allows us to finish the Proof of Theorem 2. \(\square \)

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Wang, L. Nearest neighbors estimation for long memory functional data. Stat Methods Appl 29, 709–725 (2020). https://doi.org/10.1007/s10260-019-00499-1

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Keywords

  • Asymptotic normality
  • Functional data
  • Long memory
  • Nearest neighbors estimation
  • Uniform consistency

Mathematics Subject Classification

  • 62M10