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MR3956252 62G05 62G08
Zhang, Rui (PRC-ANH-WDC) ; Wu, Yi (PRC-ANH-SM) ;Xu, Weifeng (PRC-ANH-WDC) ;Wang, Xuejun [Wang, Xue Jun1 ] (PRC-ANH-SM)
On complete consistency for the weighted estimator of nonparametric regression
models. (English summary)Rev. R. Acad. Cienc. Exactas Fı́s. Nat. Ser. A Mat. RACSAM 113 (2019), no. 3,2319–2333.
This article aims at demonstrating complete convergence [P. L. Hsü and H. E. Robbins,Proc. Nat. Acad. Sci. U.S.A. 33 (1947), 25–31; MR0019852] of a class of nonparametric
regression estimators in which regression errors obey a newly proposed dependence
structure. The regression model of interest is of the form
Yni = f (xni ) + εni ,i = 1, . . . , n,where xni are known fixed design points in a compact set A ⊂ Rd , f (·) is an unknown
regression function, and εni are widely orthant dependent (WOD) [K. Wang, Y. B. Wang
and Q. Gao, Methodol. Comput. Appl. Probab. 15 (2013), no. 1, 109–124; MR3030214]
random variables with mean zero. The weighted regression estimatorfn (x) =nXWni (x)Yni ,x ∈ A,i=1

is exclusively considered, where Wni (x) = Wni (x; xn1 , . . . , xnn ) are weight functions.
Before proceeding, definitions of complete convergence and WOD should be provided.
First, a random sequence Yn , n = 1, 2, . . . is said to converge completely to a constant
Y if∞XPr(|Yn − Y | > ) < ∞n=1

holds for every  > 0. Complete convergence is stronger than almost sure convergence inthat Yn → Y completely as n → ∞ establishes Yn → Y a.s. by the Borel-Cantelli lemma.
Second, WOD is built on two dependence structures. Random variables X1 , . . . , Xn aresaid to be widely upper orthant dependent (WUOD) if there is a finite real numbergU (n) so that( n)n\YPr(Xi > xi ) ≤ gU (n)Pr(Xi > xi )
i=1

i=1

holds for every xi ∈ (−∞, ∞), i = 1, . . . , n. In addition, X1 , . . . , Xn are said to be
widely lower orthant dependent (WLOD) if there is a finite real number gL (n) so that( n)n\YPr(Xi ≤ xi ) ≤ gL (n)Pr(Xi ≤ xi )
i=1

i=1

holds for every xi ∈ (−∞, ∞), i = 1, . . . , n. Finally, X1 , . . . , Xn are said to be WOD
if they are both WUOD and WLOD. These dependence structures were proposed
originally in the context of a stochastic risk model that can describe ruin probabilities
of insurance companies.
The main result of the article under review is that fn (x) → f (x) completely as n →∞ for every continuity point x of the function f on A if all the following regularity
conditions are satisfied:Pn(1) Pi=1 Wni (x) → 1 as n → ∞.
n(2) Pi=1 |Wni (x)| ≤ C < ∞ for every n.
n(3)i=1 |Wni (x)||f (xni ) − f (x)|1(kxni − xk > a) → 0 as n → ∞ for every a > 0.
(4) For some p ≥ 1,(a) max1≤i≤n |Wni (x)| = O(n−1/p ); and(b) E|X|2p < ∞ for a random variable X that stochastically dominates εni .
(5) g(n) = max{gU (n), gL (n)} = O(nλ ) for some λ ≥ 0.
The result is an improvement over Theorem 4.1 of X. J. Wang et al. [TEST 23 (2014),
no. 3, 607–629; MR3252097]. The theorem establishes complete convergence of fn (x)under stronger assumptions. More specifically, stronger conditions are imposed in placeof (4)(b) and (5) above, whereas all other conditions remain unchanged.
Masayuki Hirukawa
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