On the local linear modelization of the conditional density for functional and ergodic data

Abstract

In this paper, we estimate the conditional density function using the local linear approach. We treat the case when the regressor is valued in a semi-metric space, the response is a scalar and the data are observed as ergodic functional times series. Under this dependence structure, we state the almost complete consistency (a.co.) with rates of the constructed estimator. Moreover, the usefulness of our results is illustrated through their application to the conditional mode estimation.

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Proof

Proof

Preliminary technical lemmas

Firstly, we state the following technical lemmas which are needed to establish our asymptotic results.

Lemma 5

Under the assumptions (H.1),(H.3) and (H.4)(i), we have: \( \forall \left( k,l \right) \in \mathbb {N}^{*} \times \mathbb {N}\),

  1. (i)

    \(\mathbb {E}\left( K_{j}^{k} \vert \rho _{j} \vert ^{l} | \mathcal {F}_{j-1} \right) \le C h_{K}^{l} \phi _{j,x}\left( h_{K}\right) \)

  2. (ii)

    \(\mathbb {E}\left( \Gamma _{j}K_{j}| \mathcal {F}_{j-1} \right) = O \left( n h_{K}^{2} \phi _{j,x}\left( h_{K} \right) \right) \)

  3. (iii)

    \(\mathbb {E}\left( \Gamma _{1}K_{1} \right) = O \left( n h_{K}^{2} \phi _{x}\left( h_{K} \right) \right) \)

Proof

  1. (i)

    One starts by using (H.3) followed by using (H.4), we get

    $$\begin{aligned} K_{j}^{k}|\rho _{j}|^{l} h_{K}^{-l}\le & {} C K_{j}^{k} |\delta \left( X_{j}, x\right) |^{l} h_{K}^{-l} \\\le & {} C |\delta \left( X_{j}, x\right) |^{l} h_{K}^{-l} \displaystyle {\mathbb {1}_{[-1,1]}}\left( \delta (X_{j}, x)\right) , \end{aligned}$$

    and thereby, we have

    $$\begin{aligned} \mathbb {E} \left( K_{j}^{k}|\rho _{j}|^{l} h_{K}^{-l} | \mathcal {F}_{j-1}\right)\le & {} C \mathbb {P}\left( X_{j} \in B(x,h_{K}) | \mathcal {F}_{j-1}\right) , \\\le & {} C \phi _{j,x}\left( h_{K}\right) , \end{aligned}$$

    which is the claimed result.

  2. (ii)

    Recall that the fact that the kernel K is bounded on \([-1, 1]\) and under (H.3), we have

    $$\begin{aligned} |\Gamma _{j}|\le & {} n C h_{K}^{2} + n C h_{K} |\rho _{j}|. \end{aligned}$$

    So, by using (i), we find

    $$\begin{aligned} \mathbb {E}\left( \Gamma _{j}K_{j} | \mathcal {F}_{j-1} \right)\le & {} n C h_{K}^{2} \phi _{j,x}\left( h_{K}\right) + n C h_{K}^{2} \phi _{j,x}\left( h_{K} \right) \\\le & {} n C' h_{K}^{2} \phi _{j,x}\left( h_{K} \right) . \end{aligned}$$
  3. (iii)

    Combining (H.1)(iii) with part (ii) of the same lemma, and by considering \(\mathcal {F}_{j}\) as the trivial \(\sigma -\) filed, part (iii) is directly verified.

\(\square \)

Lemma 6

Under the assumptions of Lemma (5), we have

$$\begin{aligned} \displaystyle \lim _{n \rightarrow \infty } \bar{f}_{0}^{x}\left( y\right) = O(1). \end{aligned}$$

Proof

We start by applying parts (ii) and (iii) of Lemma 5 to get

$$\begin{aligned} \displaystyle \lim _{n \rightarrow \infty }\bar{f}_{0}^{x}\left( y\right)= & {} O(1) \displaystyle \lim _{n \rightarrow \infty }\frac{1}{n \phi _{x}\left( h_{K} \right) } \displaystyle \sum _{j=1}^{n} \phi _{j,x}\left( h_{K} \right) . \end{aligned}$$

Finally, we just have to use part (iii) of assumption (H.1) to obtain the claimed result. \(\square \)

Proofs of main results:

Proof of Lemma 1

Observe that

$$\begin{aligned}&\displaystyle \frac{\bar{f}^{x}_{1} \left( y\right) }{\bar{f}^{x}_{0} (y)} - f^{x} \left( y\right) \\= & {} \displaystyle \frac{1}{n h_{J}\mathbb {E} \left( \Gamma _{1} K_{1}\right) \bar{f}_{0}^{x}\left( y\right) } \displaystyle \sum _{j=1}^{n}\left\{ \mathbb {E}\left( \Gamma _{j} K_{j} J_{j} | \mathcal {F}_{j-1} \right) - h_{J} f^{x} \left( y\right) \mathbb {E}\left( \Gamma _{j} K_{j}| \mathcal {F}_{j-1}\right) \right\} \\= & {} \displaystyle \frac{1}{n h_{J}\mathbb {E} \left( \Gamma _{1} K_{1}\right) \bar{f}_{0}^{x}\left( y\right) } \displaystyle \sum _{j=1}^{n} \displaystyle \left\{ \mathbb {E}\left( \Gamma _{j} K_{j} \mathbb {E}\left( J_{j} | \mathcal {G}_{j-1}\right) | \mathcal {F}_{j-1} \right) - h_{J} f^{x} \left( y\right) \mathbb {E}\left( \Gamma _{j} K_{j}| \mathcal {F}_{j-1}\right) \right\} \\\le & {} \displaystyle \frac{1}{n h_{J}\mathbb {E} \left( \Gamma _{1} K_{1}\right) \bar{f}_{0}^{x}\left( y\right) } \displaystyle \sum _{j=1}^{n} \displaystyle \left\{ \mathbb {E}\left( \Gamma _{j} K_{j} \left| \mathbb {E}\left[ J_{j} |X_{j}\right] - h_{J} f^{x} \left( y\right) \right| | \mathcal {F}_{j-1}\right) \right\} . \end{aligned}$$

The last inequality is obtained by (H.4) (iii). Next an integration par parts and the change of variable allow to get

$$\begin{aligned} \mathbb {E} \left( J_{j} | X_{j}\right) = h_{J} \displaystyle \int _{\mathbb {R}} J \left( t\right) f^{x} \left( y- h_{J} t\right) dt, \end{aligned}$$
(6)

thus, we have

$$\begin{aligned} \left| \mathbb {E}\left[ J_{j} |X_{j}\right] - h_{J} f^{x} \left( y\right) \right| \le h_{J} \displaystyle \int _{\mathbb {R}} J \left( t\right) \left| f^{x} \left( y- h_{J} t\right) - f^{x}\left( y\right) \right| dt. \end{aligned}$$

On one side, if we use the assumption (H.2)(i) followed by (H.4) (ii) and Lemma 6, we obtain the part (4) of Lemma 1.

And on the other side, if we replace (H.2) (i) by (H.2) (ii) we obtain

$$\begin{aligned} \displaystyle {\mathbb {1}_{ B(x,h_{k})}} \left( X_{j}\right) \left| \mathbb {E}\left[ J_{j} |X_{j}\right] - h_{J} f^{x}\left( y\right) \right| \le h_{J} \displaystyle \int _{\mathbb {R}} J \left( t\right) \left( h_{K}^{b_{1}} + \left| t\right| ^{b_{2}} h_{J}^{b_{2}} \right) dt. \end{aligned}$$

Hence, we get

$$\begin{aligned} \bar{f}^{x}_{1}\left( y\right) - f^{x}\left( y\right) \bar{f}^{x}_{0} (y)= & {} \left( O\left( h_{K}^{b_{1}} \right) + O\left( h_{J}^{b_{2}} \right) \right) \times \displaystyle \frac{1}{n \mathbb {E} \left( \Gamma _{1} K_{1}\right) } \displaystyle \sum _{j=1}^{n}\mathbb {E}\left( \Gamma _{j} K_{j} | \mathcal {F}_{j-1} \right) \\= & {} \left( O\left( h_{K}^{b_{1}} \right) + O\left( h_{J}^{b_{2}} \right) \right) \times \bar{f}^{x}_{0} (y). \end{aligned}$$

Finally, making use Lemma 6 allows us to obtain the part (5) of Lemma 1.

Proof of Lemma 2

Before proving this Lemma let us start by writing that:

$$\begin{aligned} \widehat{f}^{x}_{k} (y) - \bar{f}^{x}_{k} (y)= & {} \displaystyle \frac{ 1}{n h_{J}^{k} \mathbb {E} \left( \Gamma _{1} K_{1}\right) } \displaystyle \sum _{j=1}^{n} \left( \Gamma _{j} K_{j}J_{j}^{k} - \mathbb {E}\left( \Gamma _{j} K_{j} J_{j}^{k} | \mathcal {F}_{j-1} \right) \right) \\&= :&\displaystyle \frac{1}{n h_{J}^{k}\mathbb {E} \left( \Gamma _{1} K_{1}\right) } \displaystyle \sum _{j=1}^{n} T_{j},\text{ with } \, k=0, 1, \end{aligned}$$

and where \(T_{j}\) is a triangular array of martingale differences according to the \(\sigma \)- fields\(\left( \mathcal {F}_{j-1}\right) _{j}.\) In view that \(\mathbb {E} \left( \Gamma _{j} K_{j} J_{j}^{k} | \mathcal {F}_{j-1}\right) \) is \(\mathcal {F}_{j-1}\) measurable, it follows that

$$\begin{aligned} \mathbb {E}\left( T_{j}^{2} |\mathcal {F}_{j-1}\right)= & {} \mathbb {E}\left( \left( \Gamma _{j}K_{j}\right) ^{2} J_{j}^{2 k}| \mathcal {F}_{j-1}\right) - \mathbb {E}\left( \left( \Gamma _{j}K_{j} J_{j}^{k}| \mathcal {F}_{j-1}\right) \right) ^{2} \\\le & {} \mathbb {E}\left( \left( \Gamma _{j}K_{j}\right) ^{2} \mathbb {E}\left( J_{j}^{2 k}| \mathcal {G}_{j-1}\right) | \mathcal {F}_{j-1}\right) \\\le & {} \mathbb {E}\left( \left( \Gamma _{j}K_{j}\right) ^{2} \mathbb {E}\left( J_{j}^{2k}| X_{j}\right) | \mathcal {F}_{j-1}\right) . \end{aligned}$$

Now using (6) and by assumptions (H.2)(ii) and (H.4) (ii), we get

$$\begin{aligned} \mathbb {E} \left( J_{j}^{2k} | X_{j}\right) = O\left( h_{J}^{k}\right) . \end{aligned}$$

So,

$$\begin{aligned} \mathbb {E}\left( T_{j}^{2} |\mathcal {F}_{j-1}\right) \le C h_{J}^{k} \mathbb {E}\left( \Gamma _{j}^{2} K_{j}^{2} | \mathcal {F}_{j-1}\right) . \end{aligned}$$

Thus,

$$\begin{aligned}&\mathbb {E}\left( T_{j}^{2} |\mathcal {F}_{j-1}\right) \\&\quad \le 2 C h_{J}^{k} \left( \mathbb {E}\left( \left( \displaystyle \sum _{i=1}^{n} \rho _{i}^{2} K_{i}\right) ^{2} K_{j}^{2} | \mathcal {F}_{j-1} \right) +\mathbb {E}\left( \left( \displaystyle \sum _{i=1}^{n} |\rho _{i}| K_{i} \right) ^{2}\rho _{j}^{2} K_{j}^{2} | \mathcal {F}_{j-1}\right) \right) . \\&\quad \le 2 C h_{J}^{k}\left( C n^{2} h_{K}^{4} \mathbb {E}\left( K_{j}^{2} | \mathcal {F}_{j-1} \right) + C n^{2} h_{K}^{2}\mathbb {E}\left( \rho _{j}^{2} K_{j}^{2} | \mathcal {F}_{j-1}\right) \right) . \end{aligned}$$

This last inequality is obtained under (H.3) and (H.4) (i).

Next, applying of Lemma 5 (i) allows us to get

$$\begin{aligned} \mathbb {E}\left( T_{j}^{2} |\mathcal {F}_{j-1}\right)\le & {} 2 C' n^{2} h_{J}^{k} h_{K}^{4} \phi _{j, x} \left( h_{K}\right) . \end{aligned}$$

Now, we use the exponential inequality of Lemma 1 in [21] (with \(d_{j}^{2}= C' n^{2} h_{J}^{k} h_{K}^{4} \phi _{j, x}(h_{K})\) to obtain for all \(\varepsilon > 0,\)

$$\begin{aligned} \mathbb {P}\left( |\widehat{f}^{x}_{k}(y) - \bar{f}^{x}_{k} (y)|> \varepsilon \right)= & {} \mathbb {P}\left( |\displaystyle \frac{1}{n h_{J}^{k}\mathbb {E} \left( \Gamma _{1} K_{1}\right) } \displaystyle \sum _{j=1}^{n} T_{j}| > \varepsilon \right) \\\le & {} 2 \exp \left\{ -\frac{\varepsilon ^{2} n^{2} h_{J}^{2k} \left( \mathbb {E}\left( \Gamma _{1} K_{1}\right) \right) ^{2} }{2\left( D_{n}+ C \varepsilon n h_{J}^{k}\mathbb {E}\left( \Gamma _{1} K_{1}\right) \right) }\right\} . \end{aligned}$$

Taking \( \varepsilon = \varepsilon _{0} \displaystyle \sqrt{\frac{\varphi _{x}\left( h_{K}\right) \log n}{n^{2} h_{J}^{k} \phi _{x}^{2}( h_{K} )}},\) then

$$\begin{aligned}&\mathbb {P}\left( |\widehat{f}^{x}_{k}(y) - \bar{f}^{x}_{k} (y)| > \varepsilon _{0} \displaystyle \sqrt{\frac{\varphi _{x}\left( h_{K}\right) \log n}{n^{2} h_{J}^{k} \phi _{x}^{2}( h_{K} )}} \right) \\&\quad \le 2 \exp \left\{ -\frac{ n^{2} h_{J}^{2k} \left( \mathbb {E}\left( \Gamma _{1} K_{1}\right) \right) ^{2} \varepsilon _{0}^{2} \displaystyle \frac{\varphi _{x}\left( h_{K}\right) \log n}{n^{2} h_{J}^{k} \phi _{x}^{2}( h_{K} )}}{2\left( D_{n}+ C n h_{J}^{k} \mathbb {E}\left( \Gamma _{1} K_{1}\right) \varepsilon _{0} \displaystyle \sqrt{\frac{\varphi _{x}\left( h_{K}\right) \log n}{n^{2} h_{J}^{k} \phi _{x}^{2}( h_{K} ))}}\right) }\right\} . \end{aligned}$$

Now using Lemma 5 (iii), allows us to write

$$\begin{aligned}&\mathbb {P} \left( |\widehat{f}^{x}_{k} (y) - \bar{f}^{x}_{k} (y)| > \varepsilon _{0} \sqrt{\frac{\varphi _{x}\left( h_{K}\right) \log n}{n^{2}h_{J}^{k} \phi _{x}^{2}( h_{K} )}} \right) \\&\quad \le 2 \exp \left\{ -\frac{ n^{2} h_{J}^{2k} \left( O\left( n h_{K}^{2} \phi _{x}( h_{K})\right) \right) ^{2} \varepsilon _{0}^{2} \displaystyle \frac{\varphi _{x}\left( h_{K}\right) \log n}{n^{2} h_{J}^{k} \phi _{x}^{2}( h_{K} )}}{2 n h_{J}^{k} h_{K}^{2} \varphi _{x}\left( h_{K}\right) \left( C' n h_{K}^{2}+ O \left( n \phi _{x}( h_{K}) \right) \varepsilon _{0} \displaystyle \sqrt{\frac{\log n}{n^{2} h_{J}^{k} \phi _{x}^{2}( h_{K} ) \varphi _{x}\left( h_{K}\right) }}\right) }\right\} \\&\quad \le 2 \exp \left\{ -\frac{ n^{2} h_{J}^{2k} \left( O\left( n h_{K}^{2} \phi _{x}( h_{K})\right) \right) ^{2} \varepsilon _{0}^{2} \displaystyle \frac{\varphi _{x}\left( h_{K}\right) \log n}{n^{2} h_{J}^{k} \phi _{x}^{2}( h_{K} )}}{2 n h_{J}^{k} h_{K}^{2} \varphi _{x}\left( h_{K}\right) \left( C' n h_{K}^{2}+ O(1) \varepsilon _{0} \displaystyle \sqrt{\frac{\log n}{h_{J}^{k}\varphi _{x}\left( h_{K}\right) }}\right) }\right\} . \end{aligned}$$

Now using the fact that, under (H.1) (ii) and (iii), for all n we have \(\varphi _{x}\left( h_{K}\right) \ge C n \phi _{x} ( h_{K})\) which implies that

$$\begin{aligned} \frac{\log n}{h_{J}^{k} \varphi _{x}\left( h_{K}\right) } \le C' \frac{ \varphi _{x}\left( h_{K}\right) \log n }{ n^{2} h_{J}^{k} \phi _{x}^{2}( h_{K} )}. \end{aligned}$$

Therefore, under (H.5), we have:

$$\begin{aligned} \displaystyle \lim _{n \rightarrow \infty } \frac{\log n}{ h_{J}^{k} \varphi _{x}\left( h_{K}\right) } = 0. \end{aligned}$$

It follows that

$$\begin{aligned} \mathbb {P}\left( |\widehat{f}^{x}_{k} (y) - \bar{f}^{x}_{k} (y)| > \varepsilon _{0} \sqrt{\frac{\varphi _{x}\left( h_{K}\right) \log n}{n^{2} h_{J}^{k} \phi _{x}^{2}( h_{K} )}} \right) \le 2 n^{-C_{0} \varepsilon _{0}^{2}}, \end{aligned}$$

where \(C_{0} \) is a positive constant.

Consequently, using Borel-Cantelli’s Lemma and by choosing \( \varepsilon _{0}\) large enough, we can deduce that:

$$\begin{aligned} \widehat{f}^{x}_{k} (y) - \bar{f}^{x}_{k} (y)= O_{a. co} \left( \sqrt{\frac{\varphi _{x}\left( h_{K}\right) \log n}{n^{2} h_{J}^{k} \phi _{x}^{2}( h_{K} )}} \right) . \end{aligned}$$
(7)

Finally, taking \(k=0\), this last result finish the proof of Lemma 2. \(\square \)

Proof of Lemma 3

Remarked that, under (H.1)(iii) and (H.4), we have

$$\begin{aligned} 0< \displaystyle \frac{C}{n \phi _{x}( h_{K} )} \displaystyle \sum _{j=1}^{n} \mathbb {P}\left( X_{j} \in B\left( x, h_{K}\right) | \mathcal {F}_{j-1} \right) \le \bar{f}_{0}^{x}\left( y\right) \le |\widehat{f}^{x}_{0} (y) - \bar{f}_{0}^{x}\left( y\right) | +\widehat{f}^{x}_{0} (y). \end{aligned}$$

Therefore,

$$\begin{aligned}&\mathbb {P}\left( \widehat{f}^{x}_{0} (y) \le \frac{C}{2}\right) \\&\quad \le \mathbb {P} \left( \frac{C}{ n \phi _{x}( h_{K} )} \displaystyle \sum _{j=1}^{n} \mathbb {P}\left( X_{j} \in B\left( x, h_{K}\right) | \mathcal {F}_{j-1} \right) < \frac{C}{2} + |\widehat{f}^{x}_{0} (y) - \bar{f}_{0}^{x}\left( y\right) | \right) \\&\quad \le \mathbb {P} \left( |\frac{C}{ n \phi _{x}( h_{K} )} \displaystyle \sum _{j=1}^{n} \mathbb {P}\left( X_{j} \in B\left( x, h_{K}\right) | \mathcal {F}_{j-1} \right) - |\widehat{f}^{x}_{0} (y) - \bar{f}_{0}^{x}\left( y\right) | -C| > \frac{C}{2} \right) . \end{aligned}$$

It is obvious that Lemma 2 and (H.1) (iii) allow to obtain

$$\begin{aligned} \displaystyle \sum _{n} \mathbb {P} \left( |\frac{C}{ n \phi _{x}( h_{K} )} \displaystyle \sum _{j=1}^{n} \mathbb {P}\left( X_{j} \in B\left( x, h_{K}\right) | \mathcal {F}_{j-1} \right) - |\widehat{f}^{x}_{0} (y)- \bar{f}_{0}^{x}\left( y\right) |- C| > \frac{C}{2} \right) <\infty , \end{aligned}$$

which gives the result. \(\square \)

Proof of Lemma 4

The compactness of \(\mathscr {C} \) permits us to deduce that there exists a sequence of real numbers \((y_{k})_{k=1, \ldots , d_{n}}\) such that:

$$\begin{aligned} \mathscr {C} \subset \displaystyle \bigcup _{k=1}^{d_{n}}\mathscr {C}_{k}, \, \text{ where }\, \mathscr {C}_{k} = (y_{k}-l_{n}, y_{k}+l_{n}), \end{aligned}$$

with \(l_{n}= n^{-1-\alpha }\) and \(d_{n} = O(l_{n}^{-1}).\)

We start our proof with the following decomposition:

$$\begin{aligned} \displaystyle \sup _{y \in \mathscr {C} }|\widehat{f}^{x}_{1} (y) - \bar{f}^{x}_{1} (y)|\le & {} \displaystyle \underbrace{\displaystyle \sup _{y \in \mathscr {C} }| \widehat{f}^{x}_{1} (y) - \widehat{f}^{x}_{1} (z) | } _{S_{1}}+ \displaystyle \underbrace{ \displaystyle \sup _ {y \in \mathscr {C} }| \widehat{f}^{x}_{1} (z)- \bar{f}^{x}_{1} (z)|}_{S_{2}} \\&\quad + \displaystyle \underbrace{\displaystyle \sup _{y \in \mathscr {C} }|\bar{f}^{x}_{1} (z)- \bar{f}^{x}_{1} (y)|}_{S_{3}}. \end{aligned}$$

Now, we establish the three terms.

On the one side, for the term \(S_{1}\), by using assumption (H.5), we obtain:

$$\begin{aligned} S_{1}\le & {} \displaystyle \sup _{y \in \mathscr {C} }| \frac{1}{n h_{J} \mathbb {E}\left( \Gamma _{1} K_{1}\right) } \sum _{j=1}^{n} \Gamma _{j} K_{j}| J_{j}(y) - J_{j}(z)||, \\\le & {} \displaystyle \sup _{y \in \mathscr {C} } \frac{C| y- z |}{h_{J}} \left( | \frac{1}{n h_{J} \mathbb {E}\left( \Gamma _{1} K_{1}\right) } \sum _{j=1}^{n} \Gamma _{j} K_{j}| \right) , \\\le & {} C \frac{l_{n}}{h_{J}^{2}}| \widehat{f}^{x}_{0} (y)|. \end{aligned}$$

Thus, using Lemma 3, we get :

$$\begin{aligned} S_{1} \le C \frac{l_{n}}{h_{J}^{2}}. \end{aligned}$$

Since \(l_{n}= n^{-1-\alpha }\), we obtain:

$$\begin{aligned} \frac{l_{n}}{h_{J}^{2}}= o\left( \sqrt{\frac{\varphi _{x}\left( h_{K}\right) \log n}{n^{2}h_{J}\phi _{x}^{2}( h_{K} )}}\right) . \end{aligned}$$

So, for n large enough, we find a \(\eta > 0\) such that

$$\begin{aligned} \mathbb {P}\left( S_{1} > \eta \sqrt{\frac{\varphi _{x}\left( h_{K}\right) \log n}{n^{2}h_{J}\phi _{x}^{2}( h_{K} )}}\right) =0. \end{aligned}$$
(8)

Similarly, for the term \(S_{3}\), we obtain

$$\begin{aligned} S_{3} \le C \frac{l_{n}}{h_{J}^{2}} |\bar{f}^{x}_{0} (y)|. \end{aligned}$$

Therefore, Lemma 6 allows us to write:

$$\begin{aligned} S_{3} \le C \frac{l_{n}}{h_{J}^{2}}. \end{aligned}$$

Using analogous arguments as \(S_{1}\), we can found for n large enough:

$$\begin{aligned} \mathbb {P}\left( S_{3} > \eta \sqrt{\frac{\varphi _{x}\left( h_{K}\right) \log n}{n^{2}h_{J}\phi _{x}^{2}( h_{K} )}}\right) =0. \end{aligned}$$
(9)

On the other side, to complete the proof of this Lemma, we need to prove that:

$$\begin{aligned} S_{2} = O_{a. co} \left( \sqrt{\frac{\varphi _{x}\left( h_{K}\right) \log n}{n^{2}h_{J}\phi _{x}^{2}( h_{K} )}}\right) . \end{aligned}$$

By using (7) for \(k=1\), we get for \(\eta >0\) and for all \(z \in \mathscr {C}_{k}:\)

$$\begin{aligned} \mathbb {P}\left( |\widehat{f}_{1}^{x}(z) - \bar{f}^{x}_{1}(z)| > \eta \sqrt{\frac{\varphi _{x}\left( h_{K}\right) \log n}{n^{2}h_{J}\phi _{x}^{2}(h_{K} )}} \right) \le C' n^{-C_{0} \eta ^{2}}. \end{aligned}$$

Thus, we have

$$\begin{aligned}&\mathbb {P}\left( \displaystyle \sup _{y \in \mathscr {C} } |\widehat{f}_{1}^{x}(z) - \bar{f}^{x}_{1}(z)|> \eta \sqrt{\frac{\varphi _{x}\left( h_{K}\right) \log n}{n^{2}h_{J}\phi _{x}^{2}( h_{K} )}} \right) \\&\quad \le \mathbb {P}\left( \displaystyle \max _{z \in \mathscr {C}_{k}} |\widehat{f}_{1}^{x}(z) - \bar{f}^{x}_{1}(z)|> \eta \sqrt{\frac{\varphi _{x}\left( h_{K}\right) \log n}{n^{2}h_{J}\phi _{x}^{2}( h_{K} )}} \right) \\&\quad \le 2 d_{n} \displaystyle \max _{z \in \mathscr {C}_{k}}\mathbb {P}\left( |\widehat{f}_{1}^{x}(z) - \bar{f}^{x}_{1}(z)| > \eta \sqrt{\frac{\varphi _{x}\left( h_{K}\right) \log n}{n^{2}h_{J}\phi _{x}^{2}( h_{K} )}} \right) \\&\quad \le C' n^{-C_{0} \eta ^{2} +1 + \alpha }. \end{aligned}$$

Therefore, by choosing \(\eta \) such that \(C_{0} \eta ^{2}= 2+ 2 \alpha \), we find

$$\begin{aligned} \mathbb {P}\left( \displaystyle \sup _{y \in \mathscr {C} } |\widehat{f}_{1}^{x}(z) - \bar{f}^{x}_{1}(z)| > \eta \sqrt{\frac{\varphi _{x}\left( h_{K}\right) \log n}{n^{2}h_{J}\phi _{x}^{2}( h_{K} )}} \right) \le C' n ^{-1- \alpha }. \end{aligned}$$
(10)

Finally, Lemma 4 can be deduced directly from (8), (9) and (10). \(\square \)

Proof of Corollary 1

The unimodality of \(f^{x}\) and assumption (H.6) (ii) permit us to write that \(f^{x (l)} (\Theta (x)) = f^{x (l)} (\widehat{\Theta }(x))) = 0.\) Furthermore, by a Taylor expansion of the function \(f^{x}\) at \(\Theta (x),\) we have:

$$\begin{aligned} f^{x}(\widehat{\Theta } (x)) = f^{x}(\Theta (x))+ \frac{1}{j!} f^{x (j)} (\Theta ^{*} (x)) \left( \widehat{\Theta } (x) - \Theta (x)\right) ^{j}, \end{aligned}$$
(11)

where \(\Theta ^{*} (x)\) is between \(\Theta (x) \) and \(\widehat{\Theta } (x).\)

Next, by simple manipulation we show that

$$\begin{aligned} \vert {f}^{x}(\widehat{\Theta }(x)) - f^{x}(\Theta (x)) \vert \le 2 \displaystyle \sup _{y \in \mathscr {C} }| \widehat{f}^{x} (y) - f^{x} (y) |. \end{aligned}$$
(12)

To end the proof of Corollary 1, we only need to show the following claim.

Claim

$$\begin{aligned} \displaystyle \lim _{n \rightarrow \infty } | \widehat{\Theta } (x) - \Theta (x)| = 0. \qquad a. co. \end{aligned}$$

Proof

By the continuity of the function \(f^{x}\), it follows that:

$$\begin{aligned} \forall \varepsilon>0, \exists \delta ( \varepsilon ) >0,| f^{x} (y) - f^{x}(\Theta (x)| \le \delta ( \varepsilon ) \Rightarrow | y- \Theta (x)| \le \varepsilon . \end{aligned}$$

Then, this last consideration implies that:

$$\begin{aligned} \forall \varepsilon>0, \exists \delta ( \varepsilon )>0, \mathbb {P}\left( |\widehat{\Theta } (x) - \Theta (x)|> \varepsilon \right) \le \mathbb {P}\left( |f^{x}(\widehat{\Theta } (x))- f^{x}(\Theta (x)) |> \delta ( \varepsilon ) \right) . \end{aligned}$$
(13)

Lastly, the claimed result can be deduced by combining (13) with the the statement (12) and Theorem 2. \(\square \)

Now, we return to the proof of Corollary 1.

Since \( f^{x (j)} (\Theta ^{*} (x)) \rightarrow f^{x (j)} (\Theta (x))\)   and by using (H.6)(iii), we obtain

$$\begin{aligned} \exists c >0, \, \displaystyle \sum _{n=1}^{\infty } \mathbb {P} \left( | f^{x (j)} (\Theta ^{*} (x)) |< c \right) < \infty . \end{aligned}$$
(14)

Therefore, we have

$$\begin{aligned} |\widehat{\Theta } (x) - \Theta (x)|^{j} = O\left( \displaystyle \sup _{y \in \mathscr {C} } |\widehat{f}^{x}\left( y\right) - f^{x}\left( y\right) | \right) , \quad a.co. \end{aligned}$$

We find this last result by combining the statements (11) and (12) with (14).

Finally, the proof of Corollary 1 can be easily deduced from Theorem 2. \(\square \)

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Ayad, S., Laksaci, A., Rahmani, S. et al. On the local linear modelization of the conditional density for functional and ergodic data. METRON (2020). https://doi.org/10.1007/s40300-020-00174-6

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Keywords

  • Ergodic data
  • Functional data
  • Local linear estimator
  • Conditional density
  • Nonparametric estimation
  • Conditional mode
  • Ozone concentration

Mathematics Subject Classification

  • 62G05
  • 62G08
  • 62G20
  • 62G35
  • Secondary: 62H12