Complete consistency for the estimator of nonparametric regression model based on m-END errors

Shui-Li Zhang; Tiantian Hou; Cong Qu

doi:10.1515/math-2021-0081

Article Open Access

Complete consistency for the estimator of nonparametric regression model based on m-END errors

Shui-Li Zhang , Tiantian Hou and Cong Qu

Published/Copyright: November 18, 2021

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$Open Mathematics$

From the journal Open Mathematics Volume 19 Issue 1

Abstract

In this paper, we study the complete consistency for the estimator of nonparametric regression model based on m-END errors and obtain the convergence rates of the complete consistency under more general conditions. Finally, some simulations are illustrated to verify the validity of our results.

Keywords: regression model; m-END sequence; convergence rate; complete consistency

MSC 2010: 62G05; 62G08; 60E15

1 Introduction

In many statistical models and statistical applications, the random variables are usually assumed to be independent. However, the assumption that the random variables are independent is not plausible. So many statisticians extended this condition to various dependent structures and mixing structures, such as positively associated (PA) random variables, negatively associated (NA) random variables, negatively orthant dependent (NOD) random variables, extended negatively dependent (END) random variables, ρ -mixing random variables, φ -mixing random variables, and so on.

First, let us recall the concept of END random variables, which was introduced by Liu [1] as follows.

Definition 1.1

A finite collection of random variables X 1 , X 2 , … , X n is said to be END if there exists a constant M > 0 ,

(1.1) P ( X 1 > x 1 , X 2 > x 2 , … , X n > x n ) ≤ M ∏ i = 1 n P ( X i > x i )

and

(1.2) P ( X 1 ≤ x 1 , X 2 ≤ x 2 , … , X n ≤ x n ) ≤ M ∏ i = 1 n P ( X i ≤ x i )

hold for each n ≥ 1 and all real numbers x 1 , x 2 , … , x n .

An infinite sequence { X n , n ≥ 1 } is said to be END if every finite subcollection is END. An array of random variables { X n i , 1 ≤ i ≤ n , n ≥ 1 } is called row-wise END random variables if for each n ≥ 1 , { X n i , 1 ≤ i ≤ n } is a sequence of END random variables.

Recently, inspired by the definition of m -NA random variables, Wang et al. [2] extended the concept of END random variables to the case of m -END random variables as follows.

Definition 1.2

Let m ≥ 1 be a fixed integer. A sequence of random variables { X n , n ≥ 1 } is said to be m -extended negatively dependent ( m -END) if for any n ≥ 2 and any i 1 , i 2 , … , i n such that ∣ i k − i j ∣ ≥ m for all 1 ≤ k ≠ j ≤ n , we have that X i 1 , X i 2 , … , X i n are END.

The END random variables are a very general dependent structure, which includes independent random variables, NA random variables and NOD random variables as special cases. It is easily seen that m -END random variables of END random variables (when m = 1 ). Hence, the study of limiting behavior of m -END random variables is of more interest. Since the concept of m -END random variables was introduced by Wang et al. [2], many authors were devoted to studying the limit theory. Wang et al. [2] gave the Kolmogorov exponential inequality for m -END random variables and obtained the complete convergence for arrays of m -END random variables and the complete consistency for estimator of nonparametric regression models by using the Kolmogorov exponential inequality. Wang et al. [3] studied the complete convergence and complete moment convergence for arrays of row-wise m -END random variables, and as the applications of complete convergence obtained the strong consistency of the least square estimator in the EV regression models.

Consider the following nonparametric regression model

(1.3) Y n i = f ( x n i ) + ε n i , 1 ≤ i ≤ n , n ≥ 1 ,

where { x n i } are known fixed design points from A ⊂ R q which is a given compact set for some q ≥ 1 , f ( ⋅ ) is an unknown regression function on A , and { ε n i , 1 ≤ i ≤ n } are random errors with E ε n i = 0 . As an estimator of f ( x ) , the following linear weighted estimator was given

(1.4) f n ( x ) = ∑ i = 1 n W n i ( x ) Y n i ,

where W n i ( x ) = W n i ( x , x n 1 , … , x n n ) , i = 1 , 2 , … , n are the weight functions.

Since the weighted estimator f n ( x ) was proposed by Stone [4], many authors studied the consistency and the asymptotic normality of the estimator f n ( x ) when the { ε n i , 1 ≤ i ≤ n , n ≥ 1 } are independent random variables or dependent random variables. In the case of independent samples, it goes back to Gasser and Müller [5], Georgiev and Greblicki [6], Müller [7] and others. In the case of independent samples, Wang et al. [8] studied complete moment convergence for arrays of row-wise NOD random variables and obtained the complete consistency for the estimators of nonparametric and semiparametric regression models. Zhang et al. [9] studied the complete consistency for the weighted estimator of nonparametric regression model based on widely orthant-dependent random variables and provided a numerical simulation. Asymptotic distribution of the estimator is also considered. Shen et al. [10] have given the asymptotic normality of the linear weighted estimator f n ( x ) for ρ -mixing errors.

Let us recall the concept of complete convergence of random variable sequence, which was introduced by Hsu and Robbins [11] as follows. A sequence { U n , n ≥ 1 } is said to converge completely to a constant C if

∑ n = 1 ∞ P ( ∣ U n − C ∣ > ε ) < ∞ , for all ε > 0 .

In view of the Borel-Cantelli lemma, this implies that U n → C almost surely.

The random variable arrays { X n i , 1 ≤ i ≤ n , n ≥ 1 } are stochastically dominated by random variable X if there exists a positive constant C , such that

P ( ∣ X n i ∣ > x ) ≤ C P ( ∣ X ∣ > x ) ,

for all x ≥ 0 , 1 ≤ i ≤ n , n ≥ 1 .

The main purpose of the present paper is to study the complete consistency for the estimator f n ( x ) based on the m -END errors and obtain the convergence rate of the complete consistency. The remainder of this paper is organized as follows. Section 2 states our main results. The proofs of main results and simulations are given in Sections 3 and 4, respectively. Throughout this paper, the symbols C , C 1 , C 2 , C 3 , C 4 represent positive constants whose values may change from one place to another, I ( A ) is the indicator function of set A , and ⌈ x ⌉ denotes the integer part of x .

2 Main results

2.1 Some assumptions for our results

For any function f ( x ) , we use c ( f ) to denote all continuity points of the function f on A . The norm ‖ x ‖ is the Euclidean norm. For any fixed design point x ∈ A , we state the following assumptions on weighted function W n i ( x ) , which will be used to support our main results.

∑ i = 1 n W n i ( x ) → 1 ;
∑ i = 1 n ∣ W n i ( x ) ∣ ≤ C for all n ≥ 1 , and max 1 ≤ i ≤ n ∣ W n i ( x ) ∣ = O ( n − 1 / 2 r log − s n ) , s > 1 , r > 1 / 2 ;
∑ i = 1 n ∣ W n i ( x ) ∣ ⋅ ∣ f ( x n i ) − f ( x ) ∣ I ( ‖ x n i − x ‖ > n − 1 / 2 r ) → 0 ;
∑ i = 1 n W n i ( x ) − 1 = O ( n − 1 / 2 r ) ;
∑ i = 1 n ∣ W n i ( x ) ∣ ≤ C for all n ≥ 1 , and max 1 ≤ i ≤ n ∣ W n i ( x ) ∣ = O ( n − 1 / r log − s n ) , s > 1 , r > 1 / 2 ;
∑ i = 1 n ∣ W n i ( x ) ∣ ⋅ ∣ f ( x n i ) − f ( x ) ∣ I ( ‖ x n i − x ‖ > n − 1 / 2 r ) = O ( n − 1 / 2 r ) .

The assumptions ( A 1 ) – ( A 6 ) are general, and these can be found in Wang and Si [12], Chen et al. [13], and so on.

2.2 Main results

In this subsection, we state the main results and some remarks.

Theorem 2.1

Let { ε n i , 1 ≤ i ≤ n , n ≥ 1 } be an array of row-wise m -END random variables, which is stochastically dominated by random variable X . Assume that the conditions ( A 1 ) – ( A 3 ) hold. If E ∣ X ∣ 1 + 2 r < ∞ , then for any x ∈ c ( f ) , we have

(2.1) f n ( x ) → f ( x ) , completely , as n → ∞ .

Remark 2.1

Under the moment condition E ∣ X ∣ p < ∞ for some 2 < p < 4 , and the assumptions ( A 1 ) , ( A 3 ) and

(2.2) ∑ i = 1 n ∣ W n i ( x ) ∣ ≤ C , max 1 ≤ i ≤ n ∣ W n i ( x ) ∣ = O ( n − γ ) , 2 / p < γ < 1 ,

Wang et al. [8] established the same result based on the NOD random variables. Let γ = 1 / 2 r , we can see that the conditions (2.2) are weaker than the assumption ( A 2 ) in this paper. However, note that 2 / p < γ < 1 , 2 < p < 4 , which means that 1 + 2 r ≤ 1 + p / 2 ≤ p , so our moment assumptions in Theorem 2.1 are weaker than those in Wang et al. [8]. At the same time, the m -END random variables include the NOD random variables as a special case, hence the results in Theorem 2.1 are the complement and extension to the works of Wang et al. [8].

The END random variables and m -NA random variables are the m -END random variables, so we can get the following corollaries.

Corollary 2.1

Let { ε n i , 1 ≤ i ≤ n , n ≥ 1 } be an array of row-wise END random variables, which is stochastically dominated by random variable X . Assume that the conditions ( A 1 ) – ( A 3 ) hold. If E ∣ X ∣ 1 + 2 r < ∞ , then for any x ∈ c ( f ) , we have

(2.3) f n ( x ) → f ( x ) , completely , as n → ∞ .

Remark 2.2

Under the moment condition E ∣ X ∣ 2 p < ∞ for some p ≥ 1 , the assumptions ( A 1 ) , ( A 3 ) and

(2.4) ∑ i = 1 n ∣ W n i ( x ) ∣ ≤ C , max 1 ≤ i ≤ n ∣ W n i ( x ) ∣ = O ( n − 1 / p ) ,

Wang et al. [14] established the same result based on the END random variables. Let p = 2 r , we can see that the conditions (2.4) are weaker than the assumptions ( A 2 ) . However, note that r > 1 / 2 , then 1 + 2 r ≤ 4 r = 2 p , so our moment assumptions in Corollary 2.1 are weaker than those in Wang et al. [14]. Hence, the results in Corollary 2.1 are the complements to the works of Wang et al. [14].

Corollary 2.2

Let { ε n i , 1 ≤ i ≤ n , n ≥ 1 } be an array of row-wise m -NA random variables, which is stochastically dominated by random variable X . Assume that the conditions ( A 1 ) – ( A 3 ) hold. If E ∣ X ∣ 1 + 2 r < ∞ , then for any x ∈ c ( f ) , we have

(2.5) f n ( x ) → f ( x ) , completely , as n → ∞ .

As an application of Theorem 2.1, we give the complete consistency for the nearest neighbor estimator of f ( x ) . Without loss of generality, let A = [ 0 , 1 ] , taking x n i = i n , n ≥ 1 , i = 1 , 2 , … , n . For any x ∈ A , we rewrite ∣ x n 1 − x ∣ , ∣ x n 2 − x ∣ , … , ∣ x n n − x ∣ as follows:

∣ x R 1 ( x ) ( n ) − x ∣ ≤ ∣ x R 2 ( x ) ( n ) − x ∣ ≤ ⋯ ≤ ∣ x R n ( x ) ( n ) − x ∣ .

Let 1 ≤ k n ≤ n , the nearest neighbor estimator of f ( x ) in model (1.3) is defined as follows:

f ˜ n ( x ) = ∑ i = 1 n W ˜ n i ( x ) Y n i ,

where

(2.6) W ˜ n i ( x ) = 1 / k n , if ∣ x n i − x ∣ ≤ ∣ x R k n ( x ) ( n ) − x ∣ , 0 , otherwise .

Based on the aforementioned notations, we can get the following Corollary 2.3.

Corollary 2.3

Let { ε n i , 1 ≤ i ≤ n , n ≥ 1 } be an array of row-wise m -END random variables, which is stochastically dominated by a random variable X , f ( x ) is a continuous function on A = [ 0 , 1 ] , k n = ⌈ n 1 / 2 ⌉ . If E ∣ X ∣ 4 < ∞ , then for any x ∈ [ 0 , 1 ] , we have

(2.7) f ˜ n ( x ) → f ( x ) , completely , as n → ∞ .

Theorem 2.2

Let { ε n i , 1 ≤ i ≤ n , n ≥ 1 } be an array of row-wise m -END random variables, which is stochastically dominated by random variable X . Assume that the conditions ( A 4 ) – ( A 6 ) hold. If E ∣ X ∣ 2 + 2 r < ∞ , then for any x ∈ c ( f ) , we have

(2.8) ∣ f n ( x ) − f ( x ) ∣ = O ( n − 1 / 2 r ) , completely, as n → ∞ .

Similar to Corollaries 2.1 and 2.2, we can get the following results.

Corollary 2.4

Let { ε n i , 1 ≤ i ≤ n , n ≥ 1 } be an array of row-wise END random variables, which is stochastically dominated by random variable X . Assume that the conditions ( A 4 ) – ( A 6 ) hold. If E ∣ X ∣ 2 + 2 r < ∞ , then for any x ∈ c ( f ) , we have

(2.9) ∣ f n ( x ) − f ( x ) ∣ = O ( n − 1 / 2 r ) , completely, as n → ∞ .

Remark 2.3

Let r > 1 , and

(2.10) ∑ i = 1 n W n i ( x ) − 1 = O ( n − 1 / 2 r log n ) ; ∑ i = 1 n ∣ W n i ( x ) ∣ ≤ C for all n ≥ 1 and max 1 ≤ i ≤ n ∣ W n i ( x ) ∣ = O ( n − 1 / r ) ; ∑ i = 1 n ∣ W n i ( x ) ∣ ⋅ ∣ f ( x n i ) − f ( x ) ∣ I ( ‖ x n i − x ‖ > n − 1 / 2 r log n ) = O ( n − 1 / 2 r log n ) .

For the design regression model (1.3) with END errors, under the assumptions (2.10) and moment condition E ∣ X ∣ 2 + 2 r < ∞ , Yang et al. [15] obtained the complete convergence rate for the estimator as follows.

(2.11) ∣ f n ( x ) − f ( x ) ∣ = O ( n − 1 / 2 r log n ) , completely, as n → ∞ .

Compared with the results of Yang et al. [15], the assumptions on weighted functions W n i ( x ) are different, and the complete convergence rate for the estimator is also different. So our results are the complement and extension to the works of Yang et al. [15].

Corollary 2.5

Let { ε n i , 1 ≤ i ≤ n , n ≥ 1 } be an array of row-wise m -NA random variables, which is stochastically dominated by random variable X . Assume that the conditions ( A 4 ) – ( A 6 ) hold. If E ∣ X ∣ 2 + 2 r < ∞ , then for any x ∈ c ( f ) , we have

(2.12) ∣ f n ( x ) − f ( x ) ∣ = O ( n − 1 / 2 r ) , completely , as n → ∞ .

3 Proofs of main results

Before giving the proofs of our results, we need to state some useful lemmas.

3.1 Some useful lemmas

Lemma 3.1

(Kuczmaszewska [16]) Let { X n , n ≥ 1 } be a sequence of random variables, which is stochastically dominated by a random variable X . If E ∣ X ∣ p < ∞ for some p > 0 , then for any t > 0 and n ≥ 1 , the following statements hold:

(3.1) E ∣ X n ∣ p ≤ C E ∣ X ∣ p ,

(3.2) E ∣ X n ∣ p I ( ∣ X n ∣ ≤ t ) ≤ C [ E ∣ X ∣ p I ( ∣ X ∣ ≤ t ) + t p P ( ∣ X ∣ > t ) ] ,

and

(3.3) E ∣ X n ∣ p I ( ∣ X n ∣ > t ) ≤ C E ∣ X ∣ p I ( ∣ X ∣ > t ) .

Lemma 3.2

(Wang et al. [2]) Let { X n , n ≥ 1 } be a sequence of m -END random variables. If { g n ( ⋅ ) , n ≥ 1 } are all non-decreasing (or non-increasing) functions, then { g n ( X n ) , n ≥ 1 } are still m -END random variables.

Lemma 3.3

(Wang et al. [3]) Let 1 ≤ p ≤ 2 , { X n , n ≥ 1 } be an m -END random variable sequence with E X n = 0 , E ∣ X n ∣ p < ∞ , then there exists a constant C dependent only on p and m such that

(3.4) E ∑ i = 1 n X i p ≤ C ∑ i = 1 n E ∣ X i ∣ p .

Lemma 3.4

(Yang et al. [15]) Let { X n , n ≥ 1 } be an END random variable sequence with E X n = 0 , and X n ≤ d n , a.s., n ≥ 1 , where { d n , n ≥ 1 } be a sequence of positive constants. Denote b n = max 1 ≤ i ≤ n d i , and Δ n 2 = ∑ i = 1 n E X i 2 for all n ≥ 1 . Then for any ε > 0 , there exists a constant M > 0 , such that

(3.5) P ∑ i = 1 n X i ≥ ε ≤ 2 M exp − ε 2 2 ( 2 Δ n 2 + b n ε ) .

From the definition of m -END random variables and Lemma 3.4, we can get the following result.

Lemma 3.5

Let { X n , n ≥ 1 } be m -END random variable sequences, { d n , n ≥ 1 } be a sequence of nonnegative constants. If ∣ X n ∣ ≤ d n a.s. for all n ≥ 1 , then for any r > 0 , we have

(3.6) P ∑ k = 1 n X k ≥ r ≤ 2 m M exp − r 2 2 ( 2 m 2 Δ n 2 + m r b n ) ,

where Δ n 2 = ∑ i = 1 n E X i 2 , b n = max 1 ≤ i ≤ n d i , n ≥ 1 , M is the constant in Lemma 3.4.

3.2 Proofs of main theorems

Proof of Theorem 2.1

First, it is easy to see that

(3.7) ∣ f n ( x ) − f ( x ) ∣ ≤ ∣ f n ( x ) − E f n ( x ) ∣ + ∣ E f n ( x ) − f ( x ) ∣ .

For x ∈ c ( f ) and a > 0 , then

(3.8) ∣ E f n ( x ) − f ( x ) ∣ ≤ ∑ i = 1 n ∣ W n i ( x ) ∣ ⋅ ∣ f ( x n i ) − f ( x ) ∣ I ( ‖ x n i − x ‖ ≤ a ) + ∑ i = 1 n ∣ W n i ( x ) ∣ ⋅ ∣ f ( x n i ) − f ( x ) ∣ I ( ‖ x n i − x ‖ > a ) + ∣ f ( x ) ∣ ⋅ ∑ i = 1 n W n i ( x ) − 1 .

Since x ∈ c ( f ) , for any ε > 0 , there exists a δ > 0 , such that for all y satisfying ‖ y − x ‖ < δ , we have

∣ f ( y ) − f ( x ) ∣ < ε .

For n large enough, we have that 0 < n − 1 / 2 r < δ , so

(3.9) ∣ E f n ( x ) − f ( x ) ∣ ≤ ε ∑ i = 1 n ∣ W n i ( x ) ∣ + ∣ f ( x ) ∣ ⋅ ∑ i = 1 n W n i ( x ) − 1 + ∑ i = 1 n ∣ W n i ( x ) ∣ ⋅ ∣ f ( x n i ) − f ( x ) ∣ I ( ‖ x n i − x ‖ > n − 1 / 2 r ) .

From conditions ( A 1 ) – ( A 3 ) , we can get

(3.10) ∣ E f n ( x ) − f ( x ) ∣ → 0 , for x ∈ c ( f ) .

From (3.7)–(3.10), it is enough to show

(3.11) ∣ f n ( x ) − E f n ( x ) ∣ = ∑ i = 1 n W n i ( x ) ε n i → 0 , completely, as n → ∞ .

Without loss of generality, we assume W n i ( x ) ≥ 0 . For 1 ≤ i ≤ n , n ≥ 1 , define

ε n i ( 1 ) = − n 1 / 2 r I ( ε n i < − n 1 / 2 r ) + ε n i I ( ∣ ε n i ∣ ≤ n 1 / 2 r ) + n 1 / 2 r I ( ε n i > n 1 / 2 r ) , ε n i ( 2 ) = ( ε n i + n 1 / 2 r ) I ( ε n i < − n 1 / 2 r ) + ( ε n i − n 1 / 2 r ) I ( ε n i > n 1 / 2 r ) .

Note that ε n i = ε n i ( 1 ) + ε n i ( 2 ) and E ε n i = 0 , then

(3.12) f n ( x ) − E f n ( x ) = ∑ i = 1 n W n i ( x ) ε n i = ∑ i = 1 n W n i ( x ) ( ε n i ( 1 ) − E ε n i ( 1 ) ) + ∑ i = 1 n W n i ( x ) ( ε n i ( 2 ) − E ε n i ( 2 ) ) ≕ S n 1 + S n 2 .

Since r > 1 / 2 , so E ∣ X ∣ 1 + 2 r < ∞ imply E ( X 2 ) < ∞ . From ( A 2 ) and Lemma 3.1, we have

(3.13) b n 1 = max 1 ≤ i ≤ n ∣ W n , i ( x ) ( ε n i ( 1 ) − E ε n i ( 1 ) ) ∣ ≤ C n 1 / 2 r max 1 ≤ i ≤ n ∣ W n i ( x ) ∣ ≤ C 1 log − s n ,

and

Δ n 1 2 = ∑ i = 1 n E [ W n i ( x ) ( ε n i ( 1 ) − E ε n i ( 1 ) ) ] 2 = ∑ i = 1 n W n i 2 ( x ) E ( ε n i ( 1 ) − E ε n i ( 1 ) ) 2 ≤ C max 1 ≤ i ≤ n ∣ W n i ( x ) ∣ ∑ i = 1 n ∣ W n i ( x ) ∣ E ( X 2 ) ≤ C 2 n − 1 / 2 r ( log n ) − s .

By Lemma 3.2, we can get that { ε n i ( 1 ) − E ε n i ( 1 ) , 1 ≤ i ≤ n , n ≥ 1 } is an array of row-wise m -END random variables. By Lemma 3.5, then for sufficiently large K > 0 , we have

(3.14) ∑ n = 1 ∞ P ( ∣ S n 1 ∣ ≥ K ε ) = ∑ n = 1 ∞ P ∑ i = 1 n W n i ( x ) ( ε n i ( 1 ) − E ε n i ( 1 ) ) ≥ K ε ≤ C ∑ n = 1 ∞ exp − K 2 ε 2 2 ( 2 m 2 Δ n 1 2 + m K b n 1 ε ) ≤ C ∑ n = 1 ∞ exp − K 2 ε 2 2 ( 2 C 2 m 2 n − 1 / 2 r log − s n + C 1 K m ε log − s n ) ≤ C ∑ n = 1 ∞ exp − K ε 4 C 1 m log s n ≤ C ∑ n = 1 ∞ n − 2 < ∞ .

By Markov inequality, Lemma 3.1 and ( A 2 ) , we have

(3.15) ∑ n = 1 ∞ P ( ∣ S n 2 ∣ ≥ K ε ) = ∑ n = 1 ∞ P ∑ i = 1 n W n i ( x ) ( ε n i ( 2 ) − E ε n i ( 2 ) ) ≥ K ε ≤ C ∑ n = 1 ∞ E ∑ i = 1 n W n i ( x ) ( ε n i ( 2 ) − E ε n i ( 2 ) ) ≤ C ∑ n = 1 ∞ ∑ i = 1 n ∣ W n i ( x ) ∣ E ( ∣ ε n i ∣ I ( ∣ ε n i ∣ > n 1 / 2 r ) ) ≤ C ∑ n = 1 ∞ E ( ∣ X ∣ I ( ∣ X ∣ > n 1 / 2 r ) ) ≤ C ∑ k = 1 ∞ E ( ∣ X ∣ I ( k 1 / 2 r ≤ ∣ X ∣ < ( k + 1 ) 1 / 2 r ) ) ∑ n = 1 k 1 ≤ C ∑ k = 1 ∞ k E ( ∣ X ∣ I ( k 1 / 2 r ≤ ∣ X ∣ < ( k + 1 ) 1 / 2 r ) ) ≤ C E ∣ X ∣ 1 + 2 r < ∞ .

From (3.12), (3.14), and (3.15), we have

(3.16) ∑ n = 1 ∞ P ( ∣ f n ( x ) − E f n ( x ) ∣ ≥ 2 K ε ) = ∑ n = 1 ∞ P ( ∣ S n 1 + S n 2 ∣ ≥ 2 K ε ) ≤ ∑ n = 1 ∞ P ( ∣ S n 1 ∣ ≥ K ε ) + ∑ n = 1 ∞ P ( ∣ S n 2 ∣ ≥ K ε ) < ∞ .

□

Proof of Corollary 2.3

It suffices to show the conditions of ( A 1 ) – ( A 3 ) are satisfied. By the definition of the nearest neighbor estimator and k n = ⌈ n 1 / 2 ⌉ , we can get

(3.17) ∑ i = 1 n W ˜ n i ( x ) = ∑ i = 1 k n 1 k n = 1 ,

(3.18) max 1 ≤ i ≤ n ∣ W ˜ n i ( x ) ∣ = 1 k n = 1 ⌈ n 1 / 2 ⌉ ≤ n − 1 / 3 ( log n ) − s ,

(3.19) ∑ i = 1 n ∣ W ˜ n i ( x ) ∣ ⋅ ∣ f ( x n i ) − f ( x ) ∣ I ( ∣ ∣ x n i − x ∣ ∣ ≥ n − 1 / 3 ) ≤ C ∑ i = 1 k n 1 k n ∣ ∣ x R i ( n ) ( x ) − x ∣ ∣ 2 n − 2 / 3 ≤ C ∑ i = 1 k n 1 k n ( i / n ) 2 n − 2 / 3 ≤ C 1 k n n − 2 / 3 n 2 ∑ i = 1 k n i 2 ≤ C k n 2 n 4 / 3 ≤ C n − 1 / 3 .

From (3.17)–(3.19), we can see that ( A 1 ) – ( A 3 ) hold for r = 3 / 2 .□

Proof of Theorem 2.2

In order to prove that (2.8) holds, we only need

(3.20) ∣ E f n ( x ) − f ( x ) ∣ = O ( n − 1 / 2 r ) , completely, as n → ∞ ,

and

(3.21) ∣ f n ( x ) − E f n ( x ) ∣ = O ( n − 1 / 2 r ) , completely, as n → ∞ .

Note that from assumptions ( A 4 ) – ( A 6 ) , we can see that (3.20) holds by (3.9). Since r > 1 / 2 , so E ∣ X ∣ 2 + 2 r < ∞ imply E ∣ X ∣ 2 < ∞ . From ( A 5 ) , we have

(3.22) b n 2 = max 1 ≤ i ≤ n ∣ W n i ( x ) ( ε n i ( 1 ) − E ε n i ( 1 ) ) ∣ ≤ C n 1 / 2 r max 1 ≤ i ≤ n W n i ( x ) ≤ C 3 n − 1 / 2 r log − s n ,

and

Δ n 2 2 = ∑ i = 1 n E [ W n i ( x ) ( ε n i ( 1 ) − E ε n i ( 1 ) ) ] 2 = ∑ i = 1 n W n i 2 ( x ) E ( ε n i ( 1 ) − E ε n i ( 1 ) ) 2 ≤ C max 1 ≤ i ≤ n ∣ W n i ( x ) ∣ ∑ i = 1 n ∣ W n i ( x ) ∣ E X 2 ≤ C 4 n − 1 / r ( log n ) − s .

By Lemma 3.5 and s > 1 , then

(3.23) ∑ n = 1 ∞ P ( ∣ S n 1 ∣ ≥ ε n − 1 / 2 r ) = ∑ n = 1 ∞ P ∑ i = 1 n W n i ( x ) ( ε n i ( 1 ) − E ε n i ( 1 ) ) ≥ ε n − 1 / 2 r ≤ C ∑ n = 1 ∞ exp − ε 2 n − 1 / r 2 ( 2 m 2 Δ n 2 2 + m b n 2 ε n − 1 / 2 r ) ≤ C ∑ n = 1 ∞ exp − ε 2 n − 1 / r 2 ( 2 C 4 m 2 n − 1 / r log − s n + C 3 m ε n − 1 / r log − s n ) ≤ C ∑ n = 1 ∞ exp − ε 4 C 3 m log s n ≤ C ∑ n = 1 ∞ n − 2 < ∞ .

By Markov inequality and Lemma 3.1, we have that

(3.24) ∑ n = 1 ∞ P ( ∣ S n 2 ∣ ≥ ε n − 1 / 2 r ) = ∑ n = 1 ∞ P ∑ i = 1 n W n i ( x ) ( ε n i ( 2 ) − E ε n i ( 2 ) ) ≥ ε n − 1 / 2 r ≤ C ∑ n = 1 ∞ n 1 / 2 r E ∑ i = 1 n W n i ( x ) ( ε n i ( 2 ) − E ε n i ( 2 ) ) ≤ C ∑ n = 1 ∞ n 1 / 2 r ∑ i = 1 n ∣ W n i ( x ) ∣ E ∣ ε n i ∣ I ( ∣ ε n i ∣ > n 1 / 2 r ) ≤ C ∑ n = 1 ∞ n 1 / 2 r ∑ i = 1 n ∣ W n i ( x ) ∣ E ∣ X ∣ I ( ∣ X ∣ > n 1 / 2 r ) ≤ C ∑ n = 1 ∞ n 1 / 2 r E ∣ X ∣ I ( ∣ X ∣ > n 1 / 2 r ) ≤ C ∑ n = 1 ∞ n 1 / 2 r ∑ k = n ∞ E ∣ X ∣ I ( k 1 / 2 r < ∣ X ∣ ≤ ( k + 1 ) 1 / 2 r ) ≤ C ∑ k = 1 ∞ E ∣ X ∣ I ( k 1 / 2 r < ∣ X ∣ ≤ ( k + 1 ) 1 / 2 r ) ∑ n = 1 k n 1 / 2 r ≤ C ∑ k = 1 ∞ k 1 + 1 / 2 r E ∣ X ∣ I ( k 1 / 2 r < ∣ X ∣ ≤ ( k + 1 ) 1 / 2 r ) ≤ C E ∣ X ∣ 2 r + 2 < ∞ .

From (3.12), (3.23), and (3.24), we can see that (3.21) holds.□

4 Numerical simulation

In this section, we present some simulations to show the finite sample performance of the nonparametric estimator f n ( x ) by the results of Corollary 2.3. The data are generated from model (1.3). For any n ≥ 3 , let ( ε 1 , ε 2 , … , ε n ) ∼ N n ( 0 , Σ ) , where 0 represents zero vector and

Σ = 1.16 − 0.4 0 ⋯ 0 0 0 − 0.4 1.16 − 0.4 ⋯ 0 0 0 0 − 0.4 1.16 ⋯ 0 0 0 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 0 0 0 ⋯ 1.16 − 0.4 0 0 0 0 ⋯ − 0.4 1.16 − 0.4 0 0 0 ⋯ 0 − 0.4 1.16 .

From the Joag-Dev and Proschan [17], we can get that ( ε 1 , ε 2 , … , ε n ) is an NA vector, hence it is m -END random variables. Let k n = ⌈ n 1 / 2 ⌉ , taking the sample size n as n = 100 , 500, 1000 respectively, and every points x = 0.01 , 0.02 , 0.03 , … , 0.98 , 0.99 , 1 , we use R software to compute the estimators f ˜ n ( x ) of f ( x ) with f ( x ) = exp ( x ) and f ( x ) = sin ( 2 π x ) for 1000 times, and take the mean of 1000 times as the final estimation value of f ˜ n ( x ) for each point. We obtained the comparison of f ˜ n ( x ) and f ( x ) in Figures 1, 2, 3, 4, 5 and 6. At the same time, we obtained the MSE of f ˜ n ( x ) with f ( x ) = exp ( x ) and f ( x ) = sin ( 2 π x ) at x = 0.25 , 0.45 , 0.65 , 0.85 (Table 1).

$Figure 1 Comparison of f ˜ n ( x ) {\widetilde{f}}_{n}\left(x) and f ( x ) = exp ( x ) f\left(x)=\exp \left(x) with n = 100 n=100 .$

Figure 1

Comparison of f ˜ n ( x ) and f ( x ) = exp ( x ) with n = 100 .

$Figure 2 Comparison of f ˜ n ( x ) {\widetilde{f}}_{n}\left(x) and f ( x ) = exp ( x ) f\left(x)=\exp \left(x) with n = 500 n=500 .$

Figure 2

Comparison of f ˜ n ( x ) and f ( x ) = exp ( x ) with n = 500 .

$Figure 3 Comparison of f ˜ n ( x ) {\widetilde{f}}_{n}\left(x) and f ( x ) = exp ( x ) f\left(x)=\exp \left(x) with n = 1000 n=\hspace{0.1em}\text{1000}\hspace{0.1em} .$

Figure 3

Comparison of f ˜ n ( x ) and f ( x ) = exp ( x ) with n = 1000 .

$Figure 4 Comparison of f ˜ n ( x ) {\widetilde{f}}_{n}\left(x) and f ( x ) = sin ( 2 π x ) f\left(x)=\sin \left(2\pi x) with n = 100 n=100 .$

Figure 4

Comparison of f ˜ n ( x ) and f ( x ) = sin ( 2 π x ) with n = 100 .

$Figure 5 Comparison of f ˜ n ( x ) {\widetilde{f}}_{n}\left(x) and f ( x ) = sin ( 2 π x ) f\left(x)=\sin \left(2\pi x) with n = 500 n=500 .$

Figure 5

Comparison of f ˜ n ( x ) and f ( x ) = sin ( 2 π x ) with n = 500 .

$Figure 6 Comparison of f ˜ n ( x ) {\widetilde{f}}_{n}\left(x) and f ( x ) = sin ( 2 π x ) f\left(x)=\sin \left(2\pi x) with n = 1000 n=\hspace{0.1em}\text{1000}\hspace{0.1em} .$

Figure 6

Comparison of f ˜ n ( x ) and f ( x ) = sin ( 2 π x ) with n = 1000 .

Table 1

The MSE of f n ( x )

f ( x )	x	n = 100	n = 500	n = 1000
exp ( x )	0.25	0.0504	0.0143	0.0087
	0.45	0.0617	0.0132	0.0089
	0.65	0.0375	0.0203	0.0085
	0.85	0.0353	0.0256	0.0091
sin ( 2 π x )	0.25	0.0445	0.0134	0.0093
	0.45	0.0372	0.0136	0.0093
	0.65	0.0342	0.0155	0.0085
	0.85	0.0371	0.0154	0.0089

Figures 1, 2, 3 present the comparison of f ˜ n ( x ) and f ( x ) = exp ( x ) with n = 100 , 500, 1000, and Figures 4, 5, 6 present the comparison of f ˜ n ( x ) and f ( x ) = sin ( 2 π x ) with n = 100 , 500, 1000, respectively. From Figures 1, 2, 3, 4, 5 and 6, we can see that the estimators converge to the true functions as sample sizes n increase. These do provide a numerical validation of the main results.

Acknowledgments

The authors would like to express their gratitude to the editors and the reviewers for their thoughtful comments and valuable suggestions.

Conflict of interest: Authors state no conflict of interest.

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Received: 2021-03-03

Revised: 2021-06-09

Accepted: 2021-07-05

Published Online: 2021-11-18

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

Articles in the same Issue

https://doi.org/10.1515/math-2021-0081

Keywords for this article

regression model; m-END sequence; convergence rate; complete consistency

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