Asymptotic normality of some conditional nonparametric functional parameters in high-dimensional statistics

Abstract

This paper deals with the convergence in distribution of estimators of some conditional parameters in the Functional Data Analysis framework. In fact, we consider models where the input is of functional kind and the output is a scalar. Then, we establish the asymptotic normality of the nonparametric local linear estimators of (1) the conditional distribution function and (2) the successive derivatives of the conditional density. Moreover, as by-product, we deduce the asymptotic normality of the local linear estimator of the conditional mode. Finally, to show interests of our results, on the practical point of view, we have conducted a computational study, first on a simulated data and, then on some real data concerning the forage quality.

Appendix

Proof of Lemma 6.2

By applying the Bienaymé–Tchebychev’s inequality, as $n\rightarrow \infty$, we obtain, for all $\varepsilon >0$

$$\begin{aligned} {I\!\!P}(\mid \widehat{F}^{x}_D-{\text {IE}}(\widehat{F}^{x}_D)\mid >\varepsilon )\leqslant & {} \frac{\mathrm{var}(\widehat{F}^{x}_D)}{\varepsilon ^{2}} \leqslant \varepsilon ^{-2}O\left( \frac{1}{n\phi _{x}(h_{K})}\right) \rightarrow 0. \end{aligned}$$

$\square$

Proof of Lemma 6.5

We start by writing

$$\begin{aligned} {\text {IE}}\left( \widehat{f}^{x(j)}_{N}(y)\right)= & {} \frac{n}{nh^{j+1}_{H}E(\Delta _{1}K_{1})} {\text {IE}}\left( \Delta _{1}K_{1}H^{(j+1)}_{1}\right) \nonumber \\= & {} \frac{1}{h^{j+1}_{H}{\text {IE}}\left( (n-1)W_{12}\right) } \; {\text {IE}}\left( (n-1)W_{12} \; {\text {IE}}\left( H^{(j+1)}_{1}\big |X_{1}\right) \right) , \end{aligned}$$

(20)

where

$$\begin{aligned} {\text {IE}}\left( H^{(j+1)}_{1}\big | X_{1}\right) \; = \; h^{j+1}_{H}\int _{{\mathbb {R}}} H^{(1)}(t)\; f^{X_{1}(j)} (y-h_{H}t) \; {\text {d}}t. \end{aligned}$$

Then, using a Taylor expansion, of order two, of $f^{X_{1}(j)}$ and under assumption (H2’), we obtain

$$\begin{aligned} {\text {IE}}\left( H^{(j+1)}_{1}\big |X_{1}\right) =h^{j+1}_{H}\left( f^{X_{1}(j)}(y)+ \frac{h_{\scriptstyle {H}}^2 t^2}{2}\int H^{(1)}_{1}(t){\text {d}}t\; \frac{\partial ^{j+2} f^{X_{1}}(y)}{\partial y^{j+2}} + o(h_H^2)\right) . \end{aligned}$$

Then

$$\begin{aligned} {\text {IE}}\left( \widehat{f}^{x(j)}_{N}(y)\right)= & \frac{1}{{\text {IE}}\left( W_{12}\right) } \left( {\text {IE}}\left( W_{12}\,\, f^{X_{1}(j)} (y)\right) \right. \\&\left. + \frac{h_{\scriptstyle {H}}^2 t^2}{2}\int H^{(1)}_{1}(t){\text {d}}t\ \ {\text {IE}}\left( W_{12}\frac{\partial ^{j+2} f^{X_{1}}(y)}{\partial y^{j+2}}\right) + o(h_H^2)\right) , \end{aligned}$$

and by assumption (H1’), we get

$$\begin{aligned} {\text {IE}}\left( \widehat{f}^{x(j)}_{N}(y)\right)= & {} \frac{1}{{\text {IE}}\left( W_{12}\right) } \big ( {\text {IE}}\left( W_{12}\varphi _{j}(X_{1},y)\right) \\&+ \frac{h_{\scriptstyle {H}}^2 t^2}{2}\int H^{(1)}_{1}(t){\text {d}}t \, {\text {IE}}\left( W_{12} \varphi _{j+2}(X_{1},y)\right) + o(h_H^2)\big ). \end{aligned}$$

Since ${\text {IE}}\left( \beta ^{2}_{1}W_{12}\right) =0$, from assumption (H1’) with $l\in \{j,j+2,\}$, and the fact that $\psi _{l}(0) =0$, we obtain

$$\begin{aligned}&{{\text {IE}}\left( W_{12}\varphi _{l}(X_{1},y)\right) }\\&\quad = \varphi _{l}(x,y){\text {IE}}\left( W_{12}\right) \;+\; {\text {IE}}\left( W_{12} {\text {IE}}\left( \varphi _{l}(X_{1},y) - \varphi _{l}(x,y)\big | \beta ( X_{1}, x)\right) \right) ,\\&\quad = \varphi _{l}(x, y) {\text {IE}}\left( W_{12}\right) + \frac{1}{2}\psi _{l}^{(2)} (0){\text {IE}}\left( W_{12} (\beta ( X_{1}, x)^{2})\right) +o\left( {\text {IE}}\left( W_{12} (\beta ( X_{1}, x))\right) \right) . \end{aligned}$$

Thus

$$\begin{aligned}&{{\text {IE}}\left( \widehat{f}^{x(j)}_{N}(y)\right) }\\&\quad = f^{x(j)}(y) \; +\; \frac{h_{H}^2}{2} \; \frac{\partial ^{j+2} f^{x}(y)}{\partial y^{j+2}} \; \int t^2 H^{(1)}(t) {\text {d}}t \; + \; o \left( h_{H}^{2} \; \frac{{\text {IE}}\left( \beta ( X_{1}, x) W_{12}\right) }{{\text {IE}}\left( W_{12}\right) } \right) \\&\qquad + \psi ^{(2)}_{j} (0) \; \frac{{\text {IE}}\left( \beta (X_{1}, x) W_{12}\right) }{{\text {IE}}\left( W_{12}\right) } \; +\; o \left( \frac{{\text {IE}}\left( \beta (X_{1}, x) W_{12}\right) }{{\text {IE}}\left( W_{12}\right) } \right) . \end{aligned}$$

Elsewhere, it is clear that ${\text {IE}}\left( \beta ^{2}(X_{1}, x) W_{12}\right) ={\text {IE}}\left( K_{1}\beta _{1}^{2}\right) ^{2}-{\text {IE}}\left( K_{1}\beta _{1}\right) {\text {IE}}\left( K_{1}\beta _{1}^{3}\right)$. Then, using Lemma 6.1’s result, we find

$$\begin{aligned} {\text {IE}}\left( \beta ^{2}(X_{1}, x) W_{12}\right)= & {} N(1,2)^{2}h_{K}^{4}\phi _{x}(h_{K})^{2}-o(h_{K}\phi _{x}(h_{K}))O(h_{K}^{3}\phi _{x}(h_{K}),\end{aligned}$$

(21)

$$\begin{aligned} \mathrm{and }\quad {\text {IE}}\left( W_{12}\right)= & {} N(1,2)M_{1}h_{K}^{2}\phi ^{2}_{x}(h_{K}). \end{aligned}$$

(22)

Finally, by combining (21) and (22) we obtain

$$\begin{aligned} \frac{{\text {IE}}\left( \beta ^{2} (X_{2}, x) W_{12}\right) }{{\text {IE}}\left( W_{12}\right) } \; = \; h_{K}^{2} \; \frac{N(1, 2) }{M_{1}} + o(h^{2}_{K}). \end{aligned}$$

$\square$

Proofs of Lemmas 6.3 and 6.7

We have

$$\begin{aligned}&{\frac{1}{h^{l}_{H}}{\text {IE}}\left( K^{2}_{1}\mathrm{var}\left( H^{(j+1)}\left( \frac{y-Y_{1}}{h}\right) \big |X_{1}\right) \right) }\nonumber \\&\quad =\frac{1}{h^{l}_{H}}{\text {IE}}\left( K_{1}^{2}{\text {IE}}\left( \left( H^{(j+1)}\left( \frac{y-Y_{1}}{h_{H}}\right) \right) ^{2}\big | X_{1}\right) \right) \end{aligned}$$

(23)

$$\begin{aligned}&\qquad -h^{l}_{H}{\text {IE}}\left( K_{1}^{2}{\text {IE}}^{2}\left( \frac{1}{h^{l}_{H}}H^{(j+1)}\left( \frac{y-Y_{1}}{h_{H}}\right) \big | X_{1}\right) \right) . \end{aligned}$$

(24)

Concerning the term (23), under assumptions (H5) (for $j=-1$ and $l=0$) and (H7) (for $j\ge 0$ and $l=1$) and by an integration by parts followed by a change of variable, we get

$$\begin{aligned} \frac{1}{h^{l}_{H}}{\text {IE}}\left( \left( H^{(j+1)}\left( \frac{y-Y_{1}}{h_{H}}\right) \right) ^{2}\big | X_{1}\right)= & {} \int _{{\mathbb {R}}}\left( H^{(j+1)}\left( \frac{y-z}{h_{H}}\right) \right) ^{2} f^{X_{1}}(z){\text {d}}z \\= & {} h^{l-1}_{H}\int _{{\mathbb {R}}}(H^{(j+1)}(t))^{2}f^{X_{1}}(y-th_{H}){\text {d}}t\\= & {} h^{l-1}_{H}\int _{{\mathbb {R}}}(H^{(j+1)}(t))^{2}dF^{X_{1}}(y-th_{H}){\text {d}}t. \end{aligned}$$

$\square$

Proof of Lemma 6.3

For $j=-1$ and $l=0$, by noting $H^{(0)}\left( \frac{y-Y_{1}}{h_{H}}\right) =H\left( \frac{y-Y_{1}}{h_{H}}\right)$, we have

$$\begin{aligned} {\text {IE}}\left( H^{2}\left( \frac{y-Y_{1}}{h_{H}}\right) \big | X_{1}\right)= & {} h^{-1}_{H}\int _{{\mathbb {R}}}H^{2}(t)dF^{(X_{1})}(y-th_{H})\\= & {} \int _{{\mathbb {R}}}2H(t)^{(1)}H(t)\left( F^{(X_{1})}(y-th_{H})\right) -F^{x}(y)\, {\text {d}}t\\&+\int _{{\mathbb {R}}}2H(t)^{(1)}{H(t)}F^{x}(y){\text {d}}t. \end{aligned}$$

Since $\int _{{\mathbb {R}}}2H^{(1)}(t)\, {H(t)}F^{x}(y){\text {d}}t=F^{x}(y)$, as $n\rightarrow \infty$, we deduce that

$$\begin{aligned} {\text {IE}}\left( K_{1}^{2}H^{2}\left( \frac{y-Y_{1}}{h_{H}}\right) \big | X_{1}\right) \rightarrow {\text {IE}}(K_{1}^{2})F^{x}(y), \end{aligned}$$

and

$$\begin{aligned} {\text {IE}}\left( H\left( \frac{y-Y_{1}}{h_{H}}\right) \big | X_{1}\right) -F^{x}(y)\rightarrow 0. \end{aligned}$$

So, the term (24) tends to $(F^{x}(y))^{2}$ as n tends to infinity. Then

$$\begin{aligned}{\text {IE}}\left( K_{1}^{2}{\text {IE}}^{2}\left( H\left( \frac{y-Y_{1}}{h_{H}}\right) \big | X_{1}\right) \right) \rightarrow {\text {IE}}\left( K_{1}^{2}(F^{x}(y))^{2}\right) ={\text {IE}}\left( K_{1}^{2}\right) (F^{x}(y))^{2}.\end{aligned}$$

Finally, as $n\rightarrow \infty$, we have

$$\begin{aligned} {\text {IE}}\left( K_{1}^{2}var\left( H\left( \frac{y-Y_{1}}{h}\right) \big |X_{1}\right) \right) \rightarrow {\text {IE}}(K_{1}^{2})F^{x}(y)\left( 1-F^{x}(y)\right) . \end{aligned}$$

$\square$

Proof of Lemma 6.7

For $j\ge 0$ and $l=1$, we have

$$\begin{aligned} \frac{1}{h^{l}_{H}}{\text {IE}}\left( \left( H^{(j+1)}\left( \frac{y-Y_{1}}{h_{H}}\right) \right) ^{2}\big | X_{1}\right)= & {} \int _{{\mathbb {R}}}(H^{(j+1)}(t))^{2}\left( f^{X_{1}}(y-t h_{H})\right) -f^{x}(y)\, {\text {d}}t\\&+ \int _{{\mathbb {R}}}(H^{(j+1)}(t))^{2}f^{x}(y){\text {d}}t. \end{aligned}$$

Remark that $\left( f^{X_{1}}(y-t h_{H})-f^{x}(y)\right) \rightarrow o(1) \ \mathrm{as } \ n\rightarrow \infty$. Then, as $n\rightarrow \infty$, we deduce that

$$\begin{aligned} \frac{1}{h_{H}}{\text {IE}}\left( \left( H^{(j+1)}\left( \frac{y-Y_{1}}{h_{H}}\right) \right) ^{2}\big | X_{1}\right) \rightarrow {\text {IE}}(K_{1}^{2})f^{x}(y)\int (H^{(j+1)}(t))^{2}{\text {d}}t, \end{aligned}$$

and

$$\begin{aligned} {\text {IE}}\left( H_{1}^{(j+1)}(y)\big | X_{1}\right)= & {} \int _{{\mathbb {R}}}H^{(j+1)}(h_{H}^{-1}(y-z))f^{X_{1}}(z){\text {d}}z\nonumber \\= & {} -\sum _{l=1}^{j}h^{l}_{H}\left[ H^{(j-l-1)}(h_{H}^{-1}(y-z))f^{X_{1}(l-1)}(z)\right] _{-\infty }^{+\infty }\nonumber \\&\ + h^{j}_{H}\int _{{\mathbb {R}}}H^{(1)}(h_{H}^{-1}(y-z))f^{X_{1}(j)}(z){\text {d}}z. \end{aligned}$$

(25)

So, assumption (H7) allows to cancel the first term on the right side of (25). Then, using a Taylor expansion followed by a change of variable, we obtain

$$\begin{aligned} \frac{1}{h_{H}}{\text {IE}}\left( H_{1}^{(j+1)}(y)\mid X_{1}\right) \; -\;h^{j}_{H}f^{X_{1}(j)}(y) \rightarrow 0 \quad \mathrm{as}\quad n\rightarrow \infty . \end{aligned}$$

$\square$

Proof of Lemma 6.6

We have that

$$\begin{aligned} \mathrm{var}\left( \widehat{f}^{x(j)}_{N} (y)\right)&= \frac{1}{\left( n(n-1)h^{j+1}_{H}{\text {IE}}\left( W_{12}\right) \right) ^{2}} \; \mathrm{var}\left( \displaystyle \sum _{i \ne k=1}^{n} W_{ik} H^{(j+1)}_{k}\right) \nonumber \\&= \frac{1}{\left( n(n-1)h^{j+1}_{H} {\text {IE}}\left( W_{12}\right) \right) ^{2}} \; \left( n(n-1) {\text {IE}}\left( W_{12}^{2} H_{2}^{(j+1)^{2}}\right) \right. \nonumber \\&\quad \left. + n(n-1) {\text {IE}}\left( W_{12} W_{21} H^{(j+1)}_{1} H^{(j+1)}_{2}\right) \right) \nonumber \\&\quad + n(n-1)(n-2) {\text {IE}}\left( W_{12} W_{13} H^{(j+1)}_{2} H^{(j+1)}_{3}\right) \nonumber \\&\quad + n(n-1)(n-2) {\text {IE}}\left( W_{12} W_{23} H^{(j+1)}_{2}H^{(j+1)}_{3}\right) \nonumber \\&\quad + n(n-1)(n-2) {\text {IE}}\left( W_{12} W_{31} H^{(j+1)}_{2} H^{(j+1)}_{1}\right) \nonumber \\&\quad + n(n-1)(n-2) {\text {IE}}\left( W_{12} W_{32}H_{2}^{(j+1)^{2}}\right) \nonumber \\&\quad - n(n-1)(4n-6)\left( {\text {IE}}\left( W_{12} H^{(j+1)}_{2}\right) \right) ^{2}. \end{aligned}$$

(26)

Observe that

$$\begin{aligned} \!\!\!\!\!\!\!\!\!\! {\text {IE}}\left( W_{12}H^{(j+1)}_{1}\right)&= {\text {IE}}((\beta _{1}^{2}K_{1}K_{2}-\beta _{1}\beta _{2}K_{1}K_{2})H_{1}^{(j+1)})\nonumber \\&\le {\text {IE}}(\beta _{1}^{2}K_{1})\cdot {\text {IE}}(H_{1}^{j+1}K_{1}) = O(h^{j+1}_{H}h^{2}_{K}\phi ^{2}_{x}(h_{K})). \end{aligned}$$

(27)

Then

$$\begin{aligned} \frac{1}{h^{2j+2}_{H}}\frac{{\text {IE}}\left( W_{12}H^{(j+1)}_{1}\right) ^{2}}{{\text {IE}}\left( W_{12}\right) } =\frac{1}{h^{2j+2}_{H}}\frac{ O(h^{2j+2}_{H}h^{4}_{K}\phi ^{4}_{x}(h_{K})) }{O( h^{4}_{K}\phi ^{4}_{x}(h_{K}))}=O(1). \end{aligned}$$

By some simple manipulations and using Lemma 6.1’s result, we get

$$\begin{aligned} \left\{ \begin{array}{lll} {\text {IE}}\left( W_{12}^{2} H_{2}^{(j+1)^{2}}\right) &{} = &{} O (h_{K}^{4}h_{H} \phi _{x}^{2}(h_{K})), \\ {\text {IE}}\left( W_{12} W_{21} H^{(j+1)}_{1}H^{(j+1)}_{2}\right) &{} = &{} O (h_{K}^{4}h^{2j+2}_{H}\phi _{x}^{2}(h_{K})),\\ {\text {IE}}\left( W_{12} W_{13} H^{(j+1)}_{2}H^{(j+1)}_{3}\right) &{} = &{} O(h_{K}^{4}h^{2j+2}_{H}\phi _{x}^{3}(h_{K})), \\ {\text {IE}}\left( W_{12}W_{23} H^{(j+1)}_{2} H^{(j+1)}_{3}\right) &{} = &{}O(h_{K}^{4} h^{2j+2}_{H}\phi _{x}^{3}(h_{K}))\\ {\text {IE}}\left( W_{12} W_{31} H^{(j+1)}_{2} H^{(j+1)}_{1}\right) &{} = &{}O(h_{K}^{4} h^{2j+2}_{H}\phi _{x}^{3}(h_{K}))\\ {\text {IE}}\left( W_{12} W_{32}H_{2}^{(j+1)^{2}}\right) &{} = &{}({\text {IE}}\left( \beta _{1}^{2}K_{1}\right) )^{2}{\text {IE}}\left( K_{1}^{2}H^{(j+1)^{2}}_{1}\right) \\ &{}&{}+ O (h_{K}^{4}h_{H} \phi _{x}^{3}(h_{K})),\\ {\text {IE}}\left( W_{12} H^{(j+1)}_{2}\right) ^{2} &{} = &{} O(h_{K}^{4}h^{2j+2}_{H} \phi _{x}^{2}(h_{K})). \end{array} \right. \end{aligned}$$

(28)

Therefore, the leading term in the expression of $\mathrm{var}\left( \widehat{f}^{x(j)}_{N} (y)\right)$ is

$$\begin{aligned} \frac{n(n-1)(n-2)}{(n(n-1)h^{j+1}_{H}{\text {IE}}(W_{12}))^{2}}{\text {IE}}\left( \beta _{1}^{2}K_{1}\right) ^{2}{\text {IE}}\left( K_{1}^{2}\left( H^{(j+1)}_{1}\right) ^{2}\right) . \end{aligned}$$

Combining Eqs. (26), (27) with (28), we obtain

$$\begin{aligned} \mathrm{var}\left( \widehat{f}^{x(j)}_{N}(y)\right) =\frac{{\text {IE}}\left( K_{1}^{2}\left( H^{(j+1)}_{1}\right) ^{2}\right) }{nh^{2j+2}_{H}({\text {IE}}\left( K_{1}\right) ^{2}}+o\left( \frac{1}{nh^{2j+1}_{H}\phi _{x}(h_{K})}\right) . \end{aligned}$$

(29)

Remark that

$$\begin{aligned} {\text {IE}}\left( K_{1}^{2}\left( H^{(j+1)}_{1}\right) ^{2}\right) ={\text {IE}}\left( K_{1}^{2}{\text {IE}}\left( \left( H^{(j+1)}_{1}\right) ^{2}\big |X\right) \right) . \end{aligned}$$

Hence, from Lemma 6.4’s proof, as $n\rightarrow \infty$, we obtain

$$\begin{aligned} {\text {IE}}\left( K_{1}^{2}\left( H^{(j+1)}_{1}\right) ^{2}\left( \frac{y-Y_{1}}{h_{H}}\right) \big | X_{1}\right) \rightarrow h_{H}{\text {IE}}(K_{1}^{2})f^{x}(y)\int H^{(j+1)}(t){\text {d}}t. \end{aligned}$$

(30)

Combining Eqs. (29) and (30), leads to

$$\begin{aligned} \mathrm{var}\left( \widehat{f}^{x(j)}_{N}(y)\right)= & {} \frac{f^{x}(y)\int \left( H^{(j+1)}(t)\right) ^{2}{\text {d}}t}{nh^{2j+1}_{H}\phi _{x}(h_{K})} \frac{M_{2}}{M_{1}^{2}} + o\left( \frac{1}{nh^{2j+1}_{H}\phi _{x}(h_{K})}\right) . \end{aligned}$$

$\square$