The functional kNN estimator of the conditional expectile: Uniform consistency in number of neighbors

Ibrahim M. Almanjahie; Salim Bouzebda; Zouaoui Chikr Elmezouar; Ali Laksaci

doi:10.1515/strm-2019-0029

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The functional kNN estimator of the conditional expectile: Uniform consistency in number of neighbors

Ibrahim M. Almanjahie , Salim Bouzebda , Zouaoui Chikr Elmezouar and Ali Laksaci

Published/Copyright: October 10, 2021

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From the journal Statistics & Risk Modeling Volume 38 Issue 3-4

Abstract

The main purpose of the present paper is to investigate the problem of the nonparametric estimation of the expectile regression in which the response variable is scalar while the covariate is a random function. More precisely, an estimator is constructed by using the k Nearest Neighbor procedures (kNN). The main contribution of this study is the establishment of the Uniform consistency in Number of Neighbors (UNN) of the constructed estimator. The usefulness of our result for the smoothing parameter automatic selection is discussed. Short simulation results show that the finite sample performance of the proposed estimator is satisfactory in moderate sample sizes. We finally examine the implementation of this model in practice with a real data in financial risk analysis.

Keywords: Functional data analysis; expectile regression; UNN consistency; functional nonparametric statistics; almost complete convergence rate

MSC 2010: 62G05; 62G08; 62G20; 62H12

A Mathematical developments

This section is devoted to the proofs of our results. The previously presented notation continues to be used in the following.

Proof of Theorem 3.1

Let us introduce, for ϵ 0 > 0 ,

z n = ϵ 0 ⁢ ( ϕ x - 1 ⁢ ( k 2 , n n ) min ⁡ ( k 1 , k 2 ) + log ⁡ n k 1 , n ) .

By simple analytical arguments we prove, under the first part of (A2), that there exists δ > 0 in such a way that

∑ n = 1 ∞ ℙ ( sup k 1 , n ≤ k ≤ k 2 , n | ξ p ^ ( x , t ) - ξ p ( x ) | > z n ) ≤ ∑ n = 1 ∞ ℙ ( sup k 1 , n ≤ k ≤ k 2 , n sup t ∈ [ ξ p ⁢ ( x ) - δ , ξ p ⁢ ( x ) + δ ] | G ^ ( x , t ) - G ( x , t ) | ≥ C z n ) < ∞ .

Thus, the result (3.1) is consequence of the following statement:

sup k 1 , n ≤ k ≤ k 2 , n ⁡ sup t ∈ [ ξ p ⁢ ( x ) - δ , ξ p ⁢ ( x ) + δ ] ⁡ | G ^ ⁢ ( x , t ) - G ⁢ ( x , t ) | = O a.co. ⁢ ( z n ) .

To prove this last, we follow the same ideas of [33]. Indeed, we write, for a fixed α ∈ ] 0 , 1 [ ,

ℙ { sup k 1 , n ≤ k ≤ k 2 , n sup t ∈ [ ξ p ⁢ ( x ) - δ , ξ p ⁢ ( x ) + δ ] | G ^ ( x , t ) - G ( x , t ) | ≥ z n }

≤ ℙ { sup k 1 , n ≤ k ≤ k 2 , n sup t ∈ [ ξ p ⁢ ( x ) - δ , ξ p ⁢ ( x ) + δ ] | G ^ ( x , t ) - G ( x , t ) | 𝟙 { ϕ x - 1 ( α ⁢ k 1 , n n ) ≤ h k ≤ ϕ x - 1 ( k 2 , n n ⁢ α ) } ≥ z n 2 }

+ ℙ { h ∉ ( ϕ x - 1 ( α ⁢ k 1 , n n ) , ϕ x - 1 ( k 2 , n n ⁢ α ) ) } .

It is shown in [32], that

∑ n = 1 ∞ ∑ k = k 1 , n k 2 , n ℙ ( h k ≤ ϕ x - 1 ( α ⁢ k 1 , n n ) ) < ∞ ,

∑ n = 1 ∞ ∑ k = k 1 , n k 2 , n ℙ ( h k ≥ ϕ x - 1 ( k 2 , n n ⁢ α ) ) < ∞ .

So, we have only to show that

sup ϕ x - 1 ⁢ ( α ⁢ k 1 , n n ) ≤ h ≤ ϕ x - 1 ⁢ ( k 2 , n n ⁢ α ) ⁡ sup t ∈ [ ξ p ⁢ ( x ) - δ , ξ p ⁢ ( x ) + δ ] ⁡ | G ~ ⁢ ( x , t ) - G ⁢ ( x , t ) | = O a.co. ⁢ ( z n ) ,

where

G ~ ⁢ ( x , t ) = ∑ i = 1 n K ⁢ ( h - 1 ⁢ d ⁢ ( x , X i ) ) ⁢ ( Y i - t ) ⁢ 𝟙 { ( Y i - t ) ≤ 0 } ∑ i = 1 n K ⁢ ( h - 1 ⁢ d ⁢ ( x , X i ) ) ⁢ ( Y i - t ) ⁢ 𝟙 { ( Y i - t ) > 0 } .

For this, we use the following decomposition:

G ~ ⁢ ( x , t ) - G ⁢ ( x , t ) = B ~ ⁢ ( x , t ) + D ~ ⁢ ( x , t ) G ^ D ⁢ ( x , t ) + Q ~ ⁢ ( x , t ) G ^ D ⁢ ( x , t ) ,

where

Q ~ ( x , t ) = ( G ^ N ( x , t ) - 𝔼 [ G ^ N ( x , t ) ] ) - G ( x , t ) ( G ^ D ( x , t ) - 𝔼 [ G ^ D ( x , t ) ) ,

B ~ ⁢ ( x , t ) = 𝔼 ⁢ [ G ^ N ⁢ ( x , t ) ] 𝔼 [ G ^ D ( x , t ) - G ⁢ ( x , t ) ,

D ~ ( x , t ) = - B ~ ( x , t ) ( G ^ D ( x , t ) - 𝔼 [ G ^ D ( x , t ) ) ,

with

G ^ N ⁢ ( x , t ) = 1 n ⁢ ϕ x ⁢ ( h ) ⁢ ∑ i = 1 n K ⁢ ( h - 1 ⁢ d ⁢ ( x , X i ) ) ⁢ ( Y i - t ) ⁢ 𝟙 { ( Y i - t ) ≤ 0 }

and

G ^ D ⁢ ( x , t ) = 1 n ⁢ ϕ x ⁢ ( h ) ⁢ ∑ i = 1 n K ⁢ ( h - 1 ⁢ d ⁢ ( x , X i ) ) ⁢ ( Y i - t ) ⁢ 𝟙 { ( Y i - t ) > 0 } .

The proof of Theorem 3.1 becomes a straightforward consequence of the following lemmas.

Lemma 1.

Under assumptions (A1) and (A3)–(A6), we have, as n → ∞ ,

sup a n ≤ h ≤ b n ⁡ sup t ∈ [ ξ p ⁢ ( x ) - δ , ξ p ⁢ ( x ) + δ ] ⁡ | G ^ D ⁢ ( x , t ) - 𝔼 ⁢ [ G ^ D ⁢ ( x , t ) ] | = O a.co. ⁢ ( log ⁡ n n ⁢ ϕ x ⁢ ( a n ) )

and

sup a n ≤ h ≤ b n ⁡ sup t ∈ [ ξ p ⁢ ( x ) - δ , ξ p ⁢ ( x ) + δ ] ⁡ | G ^ D ⁢ ( x , t ) - 𝔼 ⁢ [ G ^ D ⁢ ( x , t ) ] | = O a.co. ⁢ ( log ⁡ n n ⁢ ϕ x ⁢ ( a n ) ) ,

where

a n = ϕ x - 1 ⁢ ( α ⁢ k 1 , n n ) and b n = ϕ x - 1 ⁢ ( k 2 , n n ⁢ α ) .

Corollary 4.

Under the assumptions of Lemma 1, there exists a positive constant C such that

∑ n = 1 ∞ ℙ ( inf a n ≤ h ≤ b n inf t ∈ [ ξ p ⁢ ( x ) - δ , ξ p ⁢ ( x ) + δ ] G ^ D ( x , t ) < C ) < ∞ .

Lemma 2.

Under assumptions (A2), (A3) and (A5), we have, as n → ∞ ,

sup a n ≤ h ≤ b n ⁡ sup t ∈ [ ξ p ⁢ ( x ) - δ , ξ p ⁢ ( x ) + δ ] ⁡ | B ~ ⁢ ( x , t ) | = O ⁢ ( b n min ⁡ ( k 1 , k 2 ) ) .

For the sake of shortness, the proof of the intermediate results are given in brevity. The proof of the second case of Lemma 1, and Corollary 4 and Lemma 2 are omitted.

Proof of Lemma 1

We use the compactness of [ ξ p ⁢ ( x ) - δ , ξ p ⁢ ( x ) + δ ] , for which we have

[ ξ p ⁢ ( x ) - δ , ξ p ⁢ ( x ) + δ ] ⊂ ⋃ j = 1 d n ( t j - ℓ n , t j + ℓ n ) ,

with

ℓ n = n - 1 2 ⁢ ς 1 and d n = O ⁢ ( n 1 2 ⁢ ς 1 ) .

Furthermore, the monotony of 𝔼 ⁢ [ G ^ D ⁢ ( x , ⋅ ) ] and G ^ D ⁢ ( x , ⋅ ) implies that, for 1 ≤ j ≤ d n ,

𝔼 ⁢ [ G ^ D ⁢ ( x , t j - ℓ n ) ] ≤ sup t ∈ ( t j - ℓ n , t j + ℓ n ) ⁡ 𝔼 ⁢ [ G ^ D ⁢ ( x , t ) ] ≤ 𝔼 ⁢ [ G ^ D ⁢ ( x , t j + ℓ n ) ] ,

G ^ D ⁢ ( x , t j - ℓ n ) ≤ sup t ∈ ( t j - ℓ n , t j + ℓ n ) ⁡ G ^ D ⁢ ( x , t ) ≤ G ^ D ⁢ ( x , t j + ℓ n ) .

It readily follows that

sup t ∈ [ ξ p ⁢ ( x ) - δ , ξ p ⁢ ( x ) + δ ] ⁡ | G ^ D ⁢ ( x , t ) - 𝔼 ⁢ [ G ^ D ⁢ ( x , t ) ] | ≤ max 1 ≤ j ≤ d n ⁡ max z ∈ { t j - ℓ n , t j + ℓ n } ⁡ | G ^ D ⁢ ( x , z ) - 𝔼 ⁢ [ G ^ D ⁢ ( x , z ) ] | + 2 ς 1 ⁢ C 2 ⁢ ℓ n ς 1 .

Making use of the assumption (A6), we can write

ℓ n ς 1 = o ⁢ ( ( log ⁡ n n ⁢ ϕ x ⁢ ( a n ) ) 1 2 ) .

Thus, we have only to prove that

(A.1) sup a n ≤ h ≤ b n ⁡ max 1 ≤ j ≤ d n ⁡ max z ∈ { t j - ℓ n , t j + ℓ n } ⁡ | G ^ D ⁢ ( x , z ) - 𝔼 ⁢ [ G ^ D ⁢ ( x , z ) ] | = O ⁢ ( ( log ⁡ n n ⁢ ϕ x ⁢ ( a n ) ) 1 2 ) , a.co.

Notice that we have

ℙ ( sup a n ≤ h ≤ b n max 1 ≤ j ≤ d n max z ∈ { t j - ℓ n , t j + ℓ n } | G ^ D ( x , z ) - 𝔼 [ G ^ D ( x , z ) ] | > η log ⁡ n n ⁢ ϕ x ⁢ ( a n ) )

≤ 2 d n max 1 ≤ j ≤ d n max z ∈ { t j - ℓ n , t j + ℓ n } ℙ ( sup a n ≤ h ≤ b n | G ^ D ( x , z ) - 𝔼 [ G ^ D ( x , z ) ] | > η log ⁡ n n ⁢ ϕ x ⁢ ( a n ) ) .

Now, we look at the quantity

ℙ ( sup a n ≤ h ≤ b n | G ^ D ( x , z ) - 𝔼 [ G ^ D ( x , z ) ] | > η log ⁡ n n ⁢ ϕ x ⁢ ( a n ) ) for all z = t j ∓ ℓ n , 1 ≤ j ≤ d n .

The proof of the latter follows similar ideas as in [24] which are based on the Bernstein’s inequality for the empirical processes

α n ⁢ ( K ) = 1 n ⁢ ∑ i = 1 n ( K i ⁢ ( ( Y i - t ) ⁢ 𝟙 ( Y i - t s ) > 0 ) - 𝔼 ⁢ [ ( ( Y i - t s ) ⁢ 𝟙 ( Y i - t s ) > 0 ) ⁢ K i ] ) ,

where

K i = K ⁢ ( h - 1 ⁢ d ⁢ ( x , X i ) ) .

We readily infer that

ℙ { sup a n ≤ h ≤ b 0 n ⁢ ϕ x ⁢ ( h ) log ⁡ n | G ^ D ( x , z ) - 𝔼 [ G ^ D ( x , z ) ] | ≥ η 0 ′ } ≤ log ( n ) n - C ′ ⁢ η 0 2 .

An appropriate choice of η 0 permits to deduce the desired equation (A.1). Hence the proof is complete.

Acknowledgements

The authors greatly thank the Editor in Chief, an Associate Editor and an anonymous referees for a careful reading of the paper. The authors also thank and extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through the Research Groups Program under grant number R.G.P.1/64/42.

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Received: 2019-10-10

Revised: 2021-05-26

Accepted: 2021-09-28

Published Online: 2021-10-10

Published in Print: 2022-01-01

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https://doi.org/10.1515/strm-2019-0029

Keywords for this article

Functional data analysis; expectile regression; UNN consistency; functional nonparametric statistics; almost complete convergence rate