Local linear approach: Conditional density estimate for functional and censored data

Abdelkader Benkhaled; Fethi Madani

doi:10.1515/dema-2022-0018

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Local linear approach: Conditional density estimate for functional and censored data

Abdelkader Benkhaled und Fethi Madani

Veröffentlicht/Copyright: 3. August 2022

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Abstract

Let Y be a random real response, which is subject to right censoring by another random variable C . In this paper, we study the nonparametric local linear estimation of the conditional density of a scalar response variable and when the covariable takes values in a semi-metric space. Our main aim is to prove under some regularity conditions both the pointwise and the uniform almost-sure consistencies with convergence rates of the conditional density estimator related by this estimation procedure.

Keywords: functional data; local linear estimator; conditional density; small balls probability; censored data

MSC 2010: 62G05; 62G07; 62G20

1 Introduction

The investigation of the relationship between a variable of interest Y and a functional covariate X constitutes a key aspect of inference in nonparametric statistic problems, We mention [1] who presented the estimator of the conditional density when the data are functional and studied its almost complete convergence. They used the so-called local linear method.

Many interesting publications are given followed this last method. For example, in [2], the quadratic error of the local linear estimator of the conditional density was studied, while in [3], the asymptotic normality of this estimator for functional time series data were established. The uniform almost-complete convergence of the local linear conditional quantile estimator was investigated in [4]. In these papers, it is assumed that the data are fully observed.

In reality, it is not possible to observe the variable under study in a complete manner as in some fields such as reliability or survival analysis. In those cases, we observe another random variable indicating censoring. The distinguishing characteristic of censoring has attracted the attention of many researchers. Let us cite for instance [5] and [6] who introduced a nonparametric local linear estimate of the conditional density function and showed some consistency results under α -mixing dependence.

The present study extends the result of [1] to censored data, and under some regularity conditions, we establish both pointwise and uniform almost-sure consistencies with convergence rates of the conditional density estimator when the explanatory variable is of the functional type.

To this end, this article is ordered as follows. In Section 2, we recall some basic knowledge of the censored model and we construct our local linear estimator. Section 3 is devoted to prove its pointwise almost-sure convergence. Its uniform convergence is established in Section 4. Finally, in Section 5, some general conclusions will be drawn.

2 Presentation of estimator

Consider n pairs of independent random variables ( X i , Y i ) for i = 1 , … , n that we assume drawn from the pair ( X , Y ) , which is valued in ℱ × R , where ℱ is a semi-metric space equipped with a semi-metric d . In this paper we consider the problem of nonparametric estimation of the conditional density of Y given X = x when the response variable ( Y i ) for i = 1 , … , n are rightly censored. Furthermore, we denote by ( C i ) for i = 1 , … , n the censoring random variables, which are supposed independent and identically distributed with a common unknown continuous distribution function G . Thus, we observe the triplets ( X i , T i , δ i ) for i = 1 , … , n , where T i = Y i ∧ C i and δ i = 1 { Y i ≤ C i } (with ∧ denoting minimum and 1 A denoting the indicator function on a set A). We suppose that ( Y i ) for i = 1 , … , n and ( C i ) for i = 1 , … , n are independent, which ensures the identifiability of the model. The cumulative distribution function G , of the censoring random variables, is estimated by the estimator defined as follows:

G ¯ n ( y ) = ∏ i = 1 n 1 − 1 − δ ( i ) n − i + 1 1 { T ( i ) ≤ y } if y < T ( n ) , 0 otherwise ,

where T ( 1 ) < T ( 2 ) < ⋯ < T ( n ) are the order statistics of T i and δ ( i ) is the concomitant of T ( i ) , which is known to be uniformly convergent to G ¯ . Here, we adopt the fast functional locally modeling, introduced by [7] for the regression analysis, that is, we estimate the conditional density f x ( . ) by a ^ , which is obtained by minimizing the following quantity:

min ( a , b ) ∈ R 2 ∑ i = 1 n ( h H − 1 δ i G ¯ n − 1 ( T i ) H ( h H − 1 ( y − T i ) ) − a − b β ( X i , x ) ) 2 K ( h K − 1 δ ( x , X i ) ) ,

where β ( . , . ) is a known function from ℱ 2 into R such that, ∀ ξ ∈ ℱ , β ( ξ , ξ ) = 0 , with K and H are kernels and h K = h K , n (resp. h H = h H , n ) is a sequence of positive real numbers and δ ( . , . ) is chosen as a function of ℱ × ℱ such that d ( . , . ) = ∣ δ ( . , . ) ∣ . Clearly, by a simple algebra, we obtain explicitly the following definition of a ^ ≔ f ^ x ( . ) :

f ^ x ( y ) = ∑ i , j = 1 n δ i G ¯ n − 1 ( T i ) W i j ( x ) H ( h H − 1 ( y − T i ) ) h H ∑ i , j = 1 n W i j ( x ) ,

where

W i j ( x ) = β ( X i , x ) [ β ( X i , x ) − β ( X j , x ) ] K ( h K − 1 δ ( x , X i ) ) K ( h K − 1 δ ( x , X j ) )

with the convention 0 / 0 = 0 .

3 Almost-sure convergence results with rate of convergence

In what follows x denotes a fixed point in ℱ , N x denotes a fixed neighborhood of x , S R will be a fixed compact subset of R , and ϕ x ( r 1 , r 2 ) = P ( r 1 ≤ δ ( X , x ) ≤ r 2 ) .

Notice that our nonparametric model will be quite general in the sense that we will just need the following assumptions:

( H 1 ) For any r > 0 , ϕ x ( r ) ≔ ϕ x ( − r , r ) > 0 .
( H 2 ) The conditional density f x is such that: there exist b 1 > 0 , b 2 > 0 , ∀ ( y 1 , y 2 ) ∈ S R 2 , and ∀ ( x 1 , x 2 ) ∈ N x × N x
∣ f x 1 ( y 1 ) − f x 2 ( y 2 ) ∣ ≤ C x ( d b 1 ( x 1 , x 2 ) + ∣ y 1 − y 2 ∣ b 2 ) ,
where C x is a positive constant depending on x .
( H 3 ) The function β ( . , . ) is such that:
∀ x , x ′ ∈ ℱ , C 1 d ( x , x ′ ) ≤ ∣ β ( x , x ′ ) ∣ ≤ C 2 d ( x , x ′ ) , where C 1 > 0 , C 2 > 0 .
( H 4 ) The kernel K is a positive, differentiable function, which is supported within ( − 1 , 1 ) .
( H 5 ) The kernel H is a positive, bounded and Lipschitzian continuous function, satisfying:
∫ ∣ t ∣ b 2 H ( t ) d t < ∞ and ∫ H 2 ( t ) d t < ∞ .
( H 6 ) The bandwidth h K satisfies: there exists an integer n 0 , such that:
∀ n > n 0 , − 1 ϕ x ( h K ) ∫ − 1 1 ϕ x ( z h K , h K ) d d z ( z 2 K ( z ) ) d z > C 3 > 0
and
h K ∫ B ( x , h K ) β ( u , x ) d P ( u ) = o ∫ B ( x , h K ) β 2 ( u , x ) d P ( u ) ,
where B ( x , r ) = { x ′ ∈ ℱ / d ( x ′ , x ) ≤ r } and d P ( x ) is the probability measure of X .
( H 7 ) The bandwidth h H satisfies:
lim n → ∞ n γ h H = ∞ for some γ > 0 and lim n → ∞ ln n n h H ϕ x ( h K ) = 0 .

Notice that, these conditions are very standard in this context. The conditions ( H 1 ) , ( H 3 ) , and ( H 6 ) are the same as those used in [8]. While assumptions ( H 2 ) is a regularity condition that characterizes the functional space of our model and is needed to evaluate the bias term in the asymptotic results of this paper. Hypotheses ( H 5 ) and ( H 7 ) are technical conditions.

Throughout this paper, we assume that τ G < ∞ , where τ G ≔ sup { y : G ( y ) < 1 } , and let τ be a positive real number such that τ < τ G .

The following theorem gives the almost-sure convergence (a.s.) of f ^ x ( . ) .

Theorem 3.1

Under assumptions ( H 1 )–( H 7 ), we have:

sup 0 ≤ y ≤ τ ∣ f ^ x ( y ) − f x ( y ) ∣ = O ( h K b 1 + h H b 2 ) + O ln n n h H ϕ x ( h K ) , a.s.

The proof of Theorem 3.1 is a direct consequence of the decomposition:

∀ y ∈ [ 0 , τ ] ,

(1) f ^ x ( y ) − f x ( y ) = f ^ N x ( y ) − f N x ( y ) f D ( x ) + f N x ( y ) − E [ f N x ( y ) ] f D ( x ) + E [ f N x ( y ) ] − f x ( y ) f D ( x ) + f x ( y ) ( 1 − f D ( x ) ) f D ( x ) ,

where

f ^ N x ( y ) = 1 n ( n − 1 ) h H E [ W 12 ( x ) ] ∑ i ≠ j δ i G ¯ n − 1 ( T i ) W i j ( x ) H ( h H − 1 ( y − T i ) ) , f N x ( y ) = 1 n ( n − 1 ) h H E [ W 12 ( x ) ] ∑ i ≠ j δ i G ¯ − 1 ( T i ) W i j ( x ) H ( h H − 1 ( y − T i ) ) , f D ( x ) = 1 n ( n − 1 ) E [ W 12 ( x ) ] ∑ i ≠ j W i j ( x )

and of Lemmas 3.2–3.5.

Lemma 3.2

(cf. [8]) Under assumptions ( H 1 ) , ( H 3 ) , ( H 4 ) , and ( H 6 ) , we have that:^[1]

1 − f D ( x ) = O ln n n ϕ x ( h K ) , a.co.

and

∃ δ > 0 , such that ∑ i = 1 ∞ P ( f D ( x ) < δ ) < ∞ .

Lemma 3.3

Under assumptions ( H 1 ) , ( H 4 ) , ( H 6 ) , and ( H 7 ) , we have that:

sup 0 ≤ y ≤ τ ∣ f ^ N x ( y ) − f N x ( y ) ∣ = O log ( log n ) n , a.s.

Proof

∣ f ^ N x ( y ) − f N x ( y ) ∣ = 1 n ( n − 1 ) h H E [ W 12 ( x ) ] ∑ i ≠ j δ i W i j ( x ) H ( h H − 1 ( y − T i ) ) 1 G ¯ n ( T i ) − 1 G ¯ ( T i ) ≤ sup 0 ≤ y ≤ τ 1 G ¯ n ( y ) − 1 G ¯ ( y ) 1 n ( n − 1 ) h H E [ W 12 ( x ) ] ∑ i ≠ j W i j ( x ) H ( h H − 1 ( y − Y i ) ) .

Using the law of iterated logarithm (see Deheuvels and Einmah [9]) and the strong law of large numbers, we have:

sup 0 ≤ y ≤ τ 1 G ¯ n ( y ) − 1 G ¯ ( y ) = O log ( log n ) n , a.s.

and

1 n ( n − 1 ) h H E [ W 12 ( x ) ] ∑ i ≠ j W i j ( x ) H ( h H − 1 ( y − Y i ) ) ⟶ E [ h H − 1 H ( h H − 1 ( y − Y 1 ) ) ] = O ( 1 ) .

Therefore,

sup 0 ≤ y ≤ τ ∣ f ^ N x ( y ) − f N x ( y ) ∣ = O log ( log n ) n , a.s.

Lemma 3.4

Under assumptions ( H 1 ) , ( H 2 ) , and ( H 5 ) , we obtain:

sup 0 ≤ y ≤ τ ∣ E [ f N x ( y ) ] − f x ( y ) ∣ = O ( h K b 1 + h H b 2 ) .

Proof

The bias term is standard and is not affected by the dependence condition of ( X i , Y i ) . So, by the equiprobability of the couples ( X i , Y i ) , we have

∀ y ∈ [ 0 , τ ] , E [ f N x ( y ) ] = 1 h H E [ W 12 ( x ) ] E [ δ 1 G ¯ − 1 ( T 1 ) W 12 ( x ) H ( h H − 1 ( y − T 1 ) ) ] .

Using the conditional expectation properties and from the fact that

E [ δ 1 ∣ Y 1 ] = G ¯ ( Y 1 ) ,

we obtain

E [ f N x ( y ) ] = 1 h H E [ W 12 ( x ) ] E W 12 ( x ) E [ 1 G ¯ ( Y 1 ) H ( h H − 1 ( y − Y 1 ) ) E [ δ 1 ∣ Y 1 ] ∣ X 1 ] = ∫ R H ( t ) f X ( y − h H t ) d t .

Therefore,

E [ f N x ( y ) ] − f x ( y ) ∣ ≤ ∫ R H ( t ) ∣ f X ( y − h H t ) − f x ( y ) ∣ d t .

Thus, by assumption ( H 2 ) and ( H 5 ) , we obtain that:

∀ y ∈ [ 0 , τ ] , ∣ E [ f N x ( y ) ] − f x ( y ) ∣ ≤ ∫ R H ( t ) ( h K b 1 + ∣ t ∣ b 2 h H b 2 ) d t = O ( h K b 1 + h H b 2 ) ,

which finishes the proof.□

In what follows, we put, for any x ∈ ℱ , and for all i = 1 , … , n :

K i ( x ) = K ( h K − 1 δ ( x , X i ) ) , β i ( x ) = β ( X i , x ) and H i ( y ) = H ( h H − 1 ( y − T i ) ) .

Lemma 3.5

Under assumptions of Theorem 3.1, we obtain:

sup 0 ≤ y ≤ τ ∣ f N x ( y ) − E [ f N x ( y ) ] ∣ = O ln n n h H ϕ x ( h K ) , a.co.

Proof

0,τ is a compact subset of R , and it can be covered by a finite number s n of intervals of length l n at some points ( z k ) k = 1 , … , s n , i.e. [ 0 , τ ] ⊂ ⋃ k = 1 s n ( z k − l n , z k + l n ) with l n = n − 3 γ 2 − 1 2 and s n = O ( l n − 1 ) .

Let z y = arg min z ∈ { z 1 , … , z s n } ∣ y − z ∣ and consider the following decomposition:

sup y ∈ [ 0 , τ ] ∣ f N x ( y ) − E [ f N x ( y ) ] ∣ ≤ sup y ∈ [ 0 , τ ] ∣ f N x ( y ) − f N x ( z y ) ∣ ︸ A 1 + sup y ∈ [ 0 , τ ] ∣ f N x ( z y ) − E [ f N x ( z y ) ] ∣ ︸ A 2 + sup y ∈ [ 0 , τ ] ∣ E [ f N x ( z y ) ] − E [ f N x ( y ) ] ∣ ︸ A 3 .

Concerning ( A 1 ) and ( A 3 ): Under ( H 5 ) , ( H 7 ) , and Lemma 3.2, we obtain:
(2) sup y ∈ [ 0 , τ ] ∣ f N x ( y ) − f N x ( z y ) ∣ = o a.co. log n n h H ϕ x ( h K ) .
In the same way, we find:
(3) sup y ∈ [ 0 , τ ] ∣ E [ f N x ( z y ) ] − E [ f N x ( y ) ] ∣ = o a.co. log n n h H ϕ x ( h K ) .
Concerning ( A 2 ): We can write for all η > 0 :
P sup y ∈ [ 0 , τ ] ∣ f N x ( z y ) − E [ f N x ( z y ) ] ∣ > η log n n h H ϕ x ( h K ) ≤ s n max z y ∈ { z 1 , … , z s n } P ∣ f N x ( z y ) − E [ f N x ( z y ) ] ∣ > η log n n h H ϕ x ( h K ) .
All it remains to compute is the following quantity:
P ∣ f N x ( z y ) − E [ f N x ( z y ) ] ∣ > η log n n h H ϕ x ( h K ) , for all z y ∈ { z 1 , … , z s n } .
To do that, we consider the following decomposition:
f N x ( z y ) = n 2 h K 2 ϕ x 2 ( h K ) n ( n − 1 ) E [ W 12 ( x ) ] ︸ T 1 1 n ∑ j = 1 n δ j K j ( x ) H j ( z y ) G ¯ ( T j ) h H ϕ x ( h K ) ︸ T 2 1 n ∑ i = 1 n K i ( x ) β i 2 ( x ) h K 2 ϕ x ( h K ) ︸ T 3 − 1 n ∑ j = 1 n δ j K j ( x ) β j ( x ) H j ( z y ) G ¯ ( T j ) h H h K ϕ x ( h K ) ︸ T 4 1 n ∑ i = 1 n K i ( x ) β i ( x ) h K ϕ x ( h K ) ︸ T 5 ,
which implies that
f N x ( z y ) − E [ f N x ( z y ) ] = T 1 ( ( T 2 T 3 − E [ T 2 T 3 ] ) − ( T 4 T 5 − E [ T 4 T 5 ] ) ) .
Clearly,
T 2 T 3 − E [ T 2 T 3 ] = ( T 2 − E [ T 2 ] ) ( T 3 − E [ T 3 ] ) + ( T 3 − E [ T 3 ] ) E [ T 2 ] + ( T 2 − E [ T 2 ] ) E [ T 3 ] + E [ T 2 ] E [ T 3 ] − E [ T 2 T 3 ]
and
T 4 T 5 − E [ T 4 T 5 ] = ( T 4 − E [ T 4 ] ) ( T 5 − E [ T 5 ] ) + ( T 5 − E [ T 5 ] ) E [ T 4 ] + ( T 4 − E [ T 4 ] ) E [ T 5 ] + E [ T 4 ] E [ T 5 ] − E [ T 4 T 5 ] .
So, our claimed result is direct consequences of the following assertions:
(4) ∑ n s n P ∣ T i − E [ T i ] ∣ > η log n n h H ϕ x ( h K ) < ∞ , for i = 2 , 3 , 4 , 5 ,

(5) T 1 = O ( 1 ) and E [ T i ] = O ( 1 ) for i = 2 , 3 , 4 , 5 ,
and almost completely:
(6) ∣ E [ T 2 ] E [ T 3 ] − E [ T 2 T 3 ] − E [ T 4 ] E [ T 5 ] + E [ T 4 T 5 ] ∣ = o ln n n h H ϕ x ( h K ) .

Proof of (4)

For this aim, we use the Bernstein’s exponential inequality for which the main point is to evaluate asymptotically the m th order moment of:

Z i l , k = 1 h K l h H k ϕ x ( h K ) ( δ i G ¯ − k ( T i ) K i ( x ) H i k ( z y ) β i l ( x ) − E [ δ i G ¯ − k ( T i ) K i ( x ) H i k ( z y ) β i l ( x ) ] )

for l = 0 , 1 , 2 , and k = 0 , 1 .

Notice that, by the Newton’s binomial expansion and the expectation properties, we obtain:

E ∣ ( δ i G ¯ − k ( T i ) K i ( x ) H i k ( z y ) β i l ( x ) − E [ δ i G ¯ − k ( T i ) K i ( x ) H i k ( z y ) β i l ( x ) ] ) m ∣ = E ∑ d = 0 m C m d ( δ i G ¯ − k ( T i ) K i ( x ) H i k ( z y ) β i l ( x ) ) d ( E [ δ i G ¯ − k ( T i ) K i ( x ) H i k ( z y ) β i l ( x ) ] ) m − d ( − 1 ) m − d ≤ ∑ d = 0 m C m d E ∣ K 1 d ( x ) β 1 d l ( x ) E [ δ i G ¯ − d k ( T i ) H 1 d k ( z y ) ∣ X 1 ] ∣ ∣ E [ K 1 ( x ) β 1 l ( x ) E [ δ i G ¯ − k ( T i ) H 1 k ( z y ) ∣ X 1 ] ] ∣ m − d .

where C m d = m ! ∕ d ! ( m − d ) ! .

Since

E [ δ 1 G ¯ − d ( T 1 ) H 1 d ( z y ) ∣ X 1 ] = E [ E [ δ 1 G ¯ − d ( T 1 ) H 1 d ( z y ) ∣ Y 1 ] ∣ X 1 ] = h H ∫ R G ¯ 1 − d ( z y − h H t ) H 1 d ( t ) f X 1 ( z y − h H t ) d t = O ( h H ) .

Then, under (H2) and (H5), we have

E [ δ 1 G ¯ − d k ( T 1 ) H 1 d k ( z y ) ∣ X 1 ] = O ( h H k ) , for all d ≤ m and k = 0 , 1 .

Moreover, it is shown in Barrientos-Marin et al. [7] that

E [ K 1 α 1 ( x ) β 1 α 2 ( x ) ] ≤ h K α 2 ϕ x ( h K ) , for all ( α 1 , α 2 ) ∈ N ∗ × N .

Therefore,

E ∣ Z i l , k ∣ m = O ( max d ∈ { 0 , … , m } ( h H k ϕ x ( h K ) ) − d + 1 ) = O ( ( h H k ϕ x ( h K ) ) − m + 1 ) .

Finally, it suffices to use Corollary A.8-(ii) in (Ferraty and Vieu [11], page 234), first with a n = ( h H ϕ x ( h K ) ) − 1 / 2 to treat the terms T 2 and T 4 , and second with a n = ( ϕ x ( h K ) ) − 1 / 2 for the terms T 3 and T 5 . In conclusion, we obtain for all η > 0 :

P ∣ T i − E [ T i ] ∣ > η ln n n h H ϕ x ( h K ) ≤ C ′ n − C η 2 , for i = 2 , 3 , 4 , 5 .

Therefore, an appropriate choice of η permits to deduce that:

s n P ∣ T i − E [ T i ] ∣ > η ln n n h H ϕ x ( h K ) ≤ C ′ n − 1 − γ , for i = 2 , 3 , 4 , 5 . □

Proofs of (5) and (6)

Notice that, the first part of (5) has been treated in [8]. We now proceed in proving the second part of (5) and (6). For this aim, since the pairs ( X i , Y i ) , i = 1 , … , n are identically distributed, we obtain that:

E [ T 2 ] = E [ δ 1 G ¯ − 1 ( T 1 ) K 1 ( x ) H 1 ( z y ) ] h H ϕ x ( h K ) , E [ T 3 ] = E [ K 1 ( x ) β 1 2 ( x ) ] h K 2 ϕ x ( h K ) , E [ T 4 ] = E [ δ 1 G ¯ − 1 ( T 1 ) K 1 ( x ) H 1 ( z y ) β 1 ( x ) ] h K h H ϕ x ( h K ) , E [ T 5 ] = E [ K 1 ( x ) β 1 ( x ) ] h K ϕ x ( h K ) , and E [ T 2 ] E [ T 3 ] − E [ T 2 T 3 ] − E [ T 4 ] E [ T 5 ] + E [ T 4 T 5 ] = 1 − n ( n − 1 ) n 2 h K − 2 h H ϕ x ( h K ) − 2 E [ K 1 ( x ) β 1 2 ( x ) ] E [ δ 1 G ¯ − 1 ( T 1 ) K 1 ( x ) H 1 ( z y ) ] .

Thus, for both equations (5) and (6), we have to evaluate:

E [ δ 1 G ¯ − q ( T 1 ) K i ( x ) H i k ( z y ) β i l ( x ) ] , for l = 0 , 1 , 2 , and q = 0 , 1 , k = 0 , 1 .

As mentioned earlier, we condition on Y 1 and then on X 1 to show that, for all l = 0 , 1 , 2 , q = 0 , 1 , and k = 0 , 1 , we have:

E [ δ 1 G ¯ − q ( T 1 ) K i ( x ) H i k ( z y ) β i l ( x ) ] = O ( h H k E [ K i ( x ) β i l ( x ) ] ) ,

and by Lemma 3 in [8], we obtain that:

(7) E [ δ 1 G ¯ − q ( T 1 ) K i ( x ) H i k ( z y ) β i l ( x ) ] = O ( h H k h K l ϕ x ( h K ) ) .

Equality (7) leads directly to:

E [ T i ] = O ( 1 ) , for i = 2 , 3 , 4 , 5 , ∣ E [ T 2 ] E [ T 3 ] − E [ T 2 T 3 ] − E [ T 4 ] E [ T 5 ] + E [ T 4 T 5 ] ∣ = O ( h H n − 1 ) ,

which implies that:

E [ T 2 ] E [ T 3 ] − E [ T 2 T 3 ] − E [ T 4 ] E [ T 5 ] + E [ T 4 T 5 ] = O ln n n h H ϕ x ( h K ) .

Now, our lemma can be easily deduced from (2)–(6).

4 Uniform almost-sure convergence results with rate of convergence

This section is devoted to the uniform version of Theorem 3.1. More precisely, our purpose is to establish the uniform almost-sure convergence of f ^ x on some subset S ℱ of ℱ , such that: S ℱ ⊂ ⋃ k = 1 d n B ( x k , r n ) , where x k ∈ ℱ and r n (resp. d n ) is a sequence of positive real numbers. Thus, in addition to the conditions introduced in the previous section, we need the following ones.

There exists a differentiable function ϕ ( . ) , such that:
∀ x ∈ S ℱ , 0 < C ϕ ( h ) ≤ ϕ x ( h ) ≤ C ′ ϕ ( h ) < ∞ and ∃ η 0 > 0 , ∀ η < η 0 , ϕ ′ ( η ) < C ,
where C , C ′ are strictly positive constants and ϕ ′ denotes the first derivative of ϕ .
The conditional density f x satisfies, for some strictly positive constant C , and ∀ ( y 1 , y 2 ) ∈ S R × S R , ∀ ( x 1 , x 2 ) ∈ S ℱ × S ℱ :
∣ f x 1 ( y 1 ) − f x 2 ( y 2 ) ∣ ≤ C ( d b 1 ( x 1 , x 2 ) + ∣ y 1 − y 2 ∣ b 2 ) .
The function β ( . , . ) satisfies ( H 3 ) and, for some strictly positive constant C ′ , the following Lipschitz’s condition:
∀ ( x 1 , x 2 ) ∈ S ℱ × S ℱ , ∣ β ( x 1 , x ′ ) − β ( x 2 , x ′ ) ∣ ≤ C ′ d ( x 1 , x 2 ) .
The kernel K satisfies ( H 4 ) and, for some strictly positive constant C , the following Lipschitz’s condition:
∣ K ( x ) − K ( y ) ∣ ≤ C ∣ ∣ x ∣ − ∣ y ∣ ∣ .
For some γ ∈ ( 0 , 1 ) , lim n → + ∞ n γ h H = ∞ , and for r n = O ln n n , the sequence d n satisfies:
( ln n ) 2 n 1 − γ ϕ ( h K ) < ln d n < n 1 − γ ϕ ( h K ) ln n
and
∑ n = 1 ∞ n ( 3 γ + 1 ) ∕ 2 d n 1 − β < ∞ , for some β > 1 .

Notice that these hypotheses are the uniform versions of the assumed conditions in the pointwise case and have already been used in the literature.

Theorem 4.1

Under assumptions ( A 1 ) , ( A 2 ) , ( A 3 ) , ( A 4 ) , ( A 5 ) , ( H 5 ) , and ( H 6 ) , we obtain that:

sup x ∈ S ℱ sup 0 ≤ y ≤ τ ∣ f ^ x ( y ) − f x ( y ) ∣ = O ( h K b 1 ) + O ( h H b 2 ) + O a . s . ln d n n 1 − γ ϕ ( h K ) .

It is clear that, as for Theorem 3.1, Theorem 4.1’s proof can be deduced directly from the decomposition (1) and from the following intermediate results, which correspond to the uniform versions of Lemmas 3.2–3.5.

Lemma 4.2

(cf. [1]) Under assumptions ( A 1 ) , ( A 3 ) , ( A 4 ) , ( A 5 ) , and ( H 6 ) , we obtain that:

sup x ∈ S ℱ ∣ f D ( x ) − 1 ∣ = O a.co. ln d n n ϕ ( h K )

and

∑ n = 1 ∞ P inf x ∈ S ℱ f D ( x ) < 1 2 < ∞ .

Lemma 4.3

Under assumptions ( A 1 ) , ( A 3 ) , ( A 4 ) , ( A 5 ) , ( H 6 ) , and ( H 7 ) , we have that:

sup x ∈ S ℱ sup 0 ≤ y ≤ τ ∣ f ^ N x ( y ) − f N x ( y ) ∣ = O ln d n n 1 − γ ϕ ( h K ) , a.s.

Proof

∣ f ^ N x ( y ) − f N x ( y ) ∣ ≤ sup 0 ≤ y ≤ τ 1 G ¯ n ( y ) − 1 G ¯ ( y ) C h H f D ( x ) ,

then by using the law of iterated logarithm (see Deheuvels and Einmah [9]), assumption ( A 5 ) , and Lemma 4.2, we have:

sup 0 ≤ y ≤ τ 1 G ¯ n ( y ) − 1 G ¯ ( y ) = O log ( log n ) n , a.s.

and

1 h H f D ( x ) = O ln d n n 1 − γ ϕ ( h K ) , a.co .

and therefore:

sup x ∈ S ℱ sup 0 ≤ y ≤ τ ∣ f ^ N x ( y ) − f N x ( y ) ∣ = O ln d n n 1 − γ ϕ ( h K ) , a.s.

Lemma 4.4

Under the hypotheses ( A 1 ) , ( A 2 ) , and ( H 5 ) , we obtain that:

sup x ∈ S ℱ sup 0 ≤ y ≤ τ ∣ f x ( y ) − E [ f N x ( y ) ] ∣ = O ( h K b 1 ) + O ( h H b 2 ) .

Proof

It suffices to combine the proofs of Lemma 3.4, and assuming the Lipschitz’s condition uniformly on ( x , y ) ∈ S ℱ × S R .□

Lemma 4.5

Under the assumptions of Theorem 4.1, we obtain that:

sup x ∈ S ℱ sup 0 ≤ y ≤ τ ∣ f N x ( y ) − E [ f N x ( y ) ] ∣ = O a.co. ln d n n 1 − γ ϕ ( h K ) .

Proof

The proof of this lemma is based on the same decomposition’s kind as used to prove Lemma 3.5. Indeed,

f N x ( y ) = n 2 h K 2 ϕ x 2 ( h K ) n ( n − 1 ) E [ W 12 ( x ) ] ︸ S 1 1 n ∑ j = 1 n δ j K j ( x ) H j ( y ) G ¯ ( T j ) h H ϕ x ( h K ) ︸ S 2 x ( y ) 1 n ∑ i = 1 n K i ( x ) β i 2 ( x ) h K 2 ϕ x ( h K ) ︸ S 3 ( x ) − 1 n ∑ j = 1 n δ j K j ( x ) β j ( x ) H j ( y ) G ¯ ( T j ) h H h K ϕ x ( h K ) ︸ S 4 x ( y ) 1 n ∑ i = 1 n K i ( x ) β i ( x ) h K ϕ x ( h K ) ︸ S 5 ( x ) ,

and in the same fashion, all it remains to show the following uniform convergences:

(8) sup x ∈ S ℱ sup 0 ≤ y ≤ τ ∣ S i x ( y ) − E [ S i x ( y ) ] ∣ = O ln d n n 1 − γ ϕ ( h K ) , a.co. for i = 2 , 4 ,

sup x ∈ S ℱ sup 0 ≤ y ≤ τ ∣ E [ S 2 x ( y ) ] E [ S 3 ( x ) ] − E [ S 2 x ( y ) S 3 ( x ) ] − E [ S 4 x ( y ) ] E [ S 5 ( x ) ] + E [ S 4 x ( y ) S 5 ( x ) ] ∣ = O ln d n n 1 − γ ϕ ( h K ) , a.co.

and, also uniformly on x ∈ S ℱ :

S 1 = O ( 1 ) and ∣ E [ S i x ( y ) ] ∣ = O ( 1 ) , for i = 2 , 4 .

Clearly, the last two equations are direct consequences of the assumption ( A 1 ) and of Lemma 4-2 in [1]. While the proof of (8) follows the same ideas as in [12]. Indeed, by noting: j ( x ) = arg min j ∈ { 1 , 2 , … , d n } ∣ δ ( x , x j ) ∣ , z y = arg min z ∈ { z 1 , … , z s n } ∣ y − z ∣ and l n = n − 3 2 γ 1 − 1 2 , we consider the following decomposition:

∣ S i x ( y ) − E [ S i x ( y ) ] ∣ ≤ sup x ∈ S ℱ sup 0 ≤ y ≤ τ ∣ S i x ( y ) − S i x j ( x ) ( y ) ∣ ︸ E 1 i + sup x ∈ S ℱ sup 0 ≤ y ≤ τ ∣ S i x j ( x ) ( y ) − S i x j ( x ) ( z y ) ∣ ︸ E 2 i + sup x ∈ S ℱ sup 0 ≤ y ≤ τ ∣ S i x j ( x ) ( z y ) − E [ S i x j ( x ) ( z y ) ] ∣ ︸ E 3 i + sup x ∈ S ℱ sup 0 ≤ y ≤ τ ∣ E [ S i x j ( x ) ( z y ) ] − E [ S i x j ( x ) ( y ) ] ∣ ︸ E 4 i + sup x ∈ S ℱ sup 0 ≤ y ≤ τ ∣ E [ S i x j ( x ) ( y ) ] − E [ S i x ( y ) ] ∣ ︸ E 5 i .

We have, then, to evaluate each term E m i , m = 1 , 2 , 3 , 4 , 5 .

Treatment of the terms E 1 i and E 5 i . First, let us analyze the first term E 1 i for i = 2 , 4 . Since K is supported in [ − 1 , 1 ] , we can write for all k = 0 , 1 that:
E 1 i ≤ 1 n h H h K k ϕ x ( h K ) sup x ∈ S ℱ sup 0 ≤ y ≤ τ ∑ j = 1 n [ K j ( x ) β j k ( x ) − K j ( x j ( x ) ) β j k ( x j ( x ) ) ] δ j G ¯ ( T j ) H j ( y ) ≤ C n h H h K k ϕ ( h K ) G ¯ ( τ ) sup x ∈ S ℱ ∑ j = 1 n [ h K k 1 B ( x , h K ) ∩ B ( x j ( x ) , h K ) ¯ ( X i ) + k C ε 1 B ( x , h K ) ∩ B ( x j ( x ) , h K ) ( X i ) + h K k C ε 1 B ( x , h K ) ∩ B ( x j ( x ) , h K ) ( X i ) + h K k 1 B ( x j ( x ) , h K ) ∩ B ( x , h K ) ¯ ( X i ) ] .
Thus,
E 1 i ≤ sup x ∈ S ℱ ( E 11 i + E 12 i + E 13 i + E 14 i ) ,
where
E 11 i = C n h H ϕ ( h K ) G ¯ ( τ ) ∑ i = 1 n 1 B ( x , h K ) ∩ B ( x j ( x ) , h K ) ¯ ( X i ) , E 12 i = k C ε n h H h K k ϕ ( h K ) G ¯ ( τ ) ∑ i = 1 n 1 B ( x , h K ) ∩ B ( x j ( x ) , h K ) ( X i ) , E 13 i = C ε n h H ϕ ( h K ) G ¯ ( τ ) ∑ i = 1 n 1 B ( x , h K ) ∩ B ( x j ( x ) , h K ) ( X i ) , E 14 i = C n h H ϕ ( h K ) G ¯ ( τ ) ∑ i = 1 n 1 B ( x j ( x ) , h K ) ∩ B ( x , h K ) ¯ ( X i ) .
Now, to evaluate these terms E 11 i , E 12 i , E 13 i , and E 14 i , we apply a standard inequality for sums of bounded random variables (cf. Corollary A.9 in [11]) with Z i is identified such that:
Z i = 1 h H ϕ ( h K ) sup x ∈ S ℱ [ 1 B ( x , h K ) ∩ B ( x j ( x ) , h K ) ¯ ( X i ) ] for E 11 i , ε h H h K ϕ ( h K ) sup x ∈ S ℱ [ 1 B ( x , h K ) ∩ B ( x j ( x ) , h K ) ( X i ) ] for E 12 i and E 13 i , 1 h H ϕ ( h K ) sup x ∈ S ℱ [ 1 B ( x j ( x ) , h K ) ∩ B ( x , h K ) ¯ ( X i ) ] for E 14 i .
Clearly, under the second part of ( A 1 ) , we have for E 11 i and E 14 i :
Z 1 = O 1 h H ϕ ( h K ) , E [ Z 1 ] = O ε h H ϕ ( h K ) and var ( Z 1 ) = O ε h H 2 ( ϕ ( h K ) ) 2 .
So that, we obtain:
E 11 i = O ε h H ϕ ( h K ) + O a.co. ε ln n n h H 2 ϕ ( h K ) 2 .
In the same way, assumption ( A 5 ) allows to obtain, for E 12 i or E 13 i case,
Z 1 = O ε h H h K ϕ ( h K ) , E [ Z 1 ] = O ε h H h K and var ( Z 1 ) = O ε 2 h H 2 h k 2 ϕ ( h K ) ,
which implies that:
E 12 i = O a.co. ln d n n 1 − γ ϕ ( h K ) .
To achieve the study of the term E 1 i , it suffices to put together all the intermediate results and to use ( A 5 ) to obtain:
(9) E 1 i = O a.co. ln d n n 1 − γ ϕ ( h K ) .
Furthermore, since:
E 5 i ≤ E [ sup x ∈ S ℱ sup 0 ≤ y ≤ τ ∣ S i x j ( x ) ( y ) − S i x ( y ) ∣ ] ,
we have also:
(10) E 5 i = O ln d n n 1 − γ ϕ ( h K ) .
Treatment of the terms E 2 i and E 4 i . By using the Lipschitz’s condition on the kernel H , one can write:
∣ S i x j ( x ) ( y ) − S i x j ( x ) ( z y ) ∣ ≤ C 1 n G ¯ ( τ ) h K k h H ϕ ( h K ) ∑ i ′ = 1 n K i ′ ( x j ( x ) ) β i ′ k ( x j ( x ) ) ∣ H i ′ ( y ) − H i ′ ( z y ) ∣ ≤ C ′ l n h H 2 S i ( x j ( x ) ) ,
where S i ( ⋅ ) for i = 2 , 4 , are defined and treated (cf. Lemma 4-2 in [1]). Thus, by using the facts that: lim n → ∞ n γ h H = ∞ and l n = n − 3 2 γ − 1 2 , we obtain:
(11) E 2 i = O a.co. ln d n n 1 − γ ϕ ( h K ) and E 4 i = O a.co. ln d n n 1 − γ ϕ ( h K ) .
Treatment of the term E 3 i . For all η > 0 , we have that:
P E 3 i > η ln d n n h H ϕ ( h K ) ≤ s n d n max i ′ ∈ { 1 , 2 , … , s n } max j ′ ∈ { 1 , … , d n } P ∣ S i x j ′ ( z i ′ ) − E [ S i x j ′ ( z i ′ ) ] ∣ > η ln d n n h H ϕ ( h K ) .
This last probability can be treated by using the classical Bernstein’s inequality, with a n = ( h H ϕ x ( h K ) ) − 1 / 2 . Recall that, the choice of a n is motivated by moment of order m of Z i l , k , q computed in Lemma 3.5’s proof. That allows, finally, to:
∀ i ′ ≤ s n , P ∣ S i x j ′ ( z i ′ ) − E [ S i x j ′ ( z i ′ ) ] ∣ > η ln d n n h H ϕ ( h K ) ≤ 2 exp { − C η 2 ln d n } .
Therefore, since s n = O ( l n − 1 ) = O n 3 2 γ + 1 2 , and by choosing C η 2 = β , one obtain:
s n d n max i ′ ∈ { 1 , 2 , … , s n } max j ′ ∈ { 1 , … , d n } P ∣ S i x j ′ ( z i ′ ) − E [ S i x j ′ ( z i ′ ) ] ∣ > η ln d n n h H ϕ ( h K ) ≤ C ′ s n d n 1 − C η 2 .
By using the fact that lim n → ∞ n γ h H = ∞ and the second part of condition ( A 5 ) , one obtains:
(12) E 3 i = O a.co. ln d n n 1 − γ ϕ ( h K ) .
Thus, Lemma 4.5’s result can be easily deduced from (9), (10), (11), and (12).□

5 Conclusion

This paper proposes the functional local linear modeling for the conditional density alternative to the kernel method to overcome the boundary problems created by the latter, when the variable of interest is subjected to random right censoring, the most frequent case of censorship. In practice, the proposed local linear estimation is easily exploitable, despite the various parameters intervening in its definition. This difficulty was overcome by employing same procedures as those used in the classical kernel method for choosing these parameters.

Moreover, we mention that our asymptotic study can be also considered as a preliminary investigation allowing the opening of several theoretical problems. Among these problems, we can cite: the conditional hazard estimation, the local linear estimation of the conditional expected shortfall, the generalization of our results on the method by local polynomials of order greater than 1, and the k nearest neighbors local linear estimator of functional conditional density.

Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: On behalf of all authors, the corresponding author states that there is no conflict of interest.
Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

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Received: 2021-06-10

Revised: 2022-01-28

Accepted: 2022-04-12

Published Online: 2022-08-03

This work is licensed under the Creative Commons Attribution 4.0 International License.

Artikel in diesem Heft

https://doi.org/10.1515/dema-2022-0018

Schlagwörter für diesen Artikel

functional data; local linear estimator; conditional density; small balls probability; censored data

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