# The Rate of Strong Consistency of the Nearest Neighbor Density Estimator for Negatively Dependent Random Variables

Authors: Xiang Zeng; Qunying Wu
Journal: ISRN Applied Mathematics
 (2012)

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## Abstract

Let{Xn:n≥1} be a sequence of negatively dependent random variables. Based on (X1,…,Xn), in this paper we investigate the rate of pointwise consistency and strong
consistency of the nonparametric density estimator proposed by Yu. We extend the correspondent results under the negatively dependent samples.


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## Body

## 1. Introduction and Lemmas

In Yu [1] a very simple and useful nonparametric estimate of a density based on a sample on R has been defined. If mn is an integer, the nonparametric estimate fn(x) of f(x) is defined by dividing 2mn by n times the length of the smallest interval containing x which consists of 2mn of the n observations in which half of them lie on the left side of x and half on the right side. Formally,
(1.1)fn(x)={2mnn(X(2mn+j)-X(j+1))ifx∈[X(mn+j),X(mn+j+1)),forj=0,1,…,n-2mn,0ifx<X(mn)orx≥X(n-mn+1),
where (X(1),…,X(n)) is the order statistic of (X1,…,Xn). In Yu [1] It is showed that in the case of independent and identically distributed observations fn(x) converges to f(x) in probability for almost all x and converges to f(x) with probability one for almost all x. Moreover, the strong uniform consistency of this estimator for f uniformly continuous was also established.Since Bozorgnia et al. [2] studied sequences of negatively dependent random variables, many statisticians have investigated this topic with interest. A series of useful results have been established. One can refer to Amini and Bozorgnia [3] for complete convergence for negatively dependent random variables, Fakoor and Azarnoosh [4] for probability inequalities for sums of negatively dependent random variables, Liu and Zhang [5] for the consistency and asymptotic normality of nearest neighbor density estimator under β-mixing condition, Wu [6] for complete convergence for weighted sums of sequences of negatively dependent random variables, and Wu and Jiang [7] for the strong consistency of estimator in linear model for negatively dependent random samples. From (1.1), we investigate the rate of consistency of the negatively dependent samples.The following set of definitions and Lemmas will be needed.Definition 1.1.
Random variablesX and Y are said to be negatively dependent (ND) if
(1.2)P(X≤x,Y≤y)≤P(X≤x)P(Y≤y),
for all x,y∈R. A collection of random variables is said to be pairwise negatively dependent (PND) if every pair of random variables in the collection satisfies (1.2).It is important to note that (1.2) implies
(1.3)P(X>x,Y>y)≤P(X>x)P(Y>y),
for all x,y∈R. Moreover, it follows that (1.3) implies (1.2), and hence, (1.2) and (1.3) are equivalent. Ebrahimi and Ghosh [8] showed that (1.2) and (1.3) are not equivalent for a collection of three or more random variables. Consequently, the following definition is needed to define sequences of negatively dependent random variables.Definition 1.2 (see [2]).
The random variablesX1,…,Xn are said to be negatively dependent (ND) if for all real x1,…,xn,
(1.4)P(⋂j=1n‍(Xj≤xj))≤∏j=1n‍P(Xj≤xj),P(⋂j=1n‍(Xj>xj))≤∏j=1n‍P(Xj>xj).
An infinite random sequence {Xn:n≥1} is said to be ND if every finite subset X1,…,Xn is ND.Lemma 1.3 (see [2]).
LetX1,…,Xn be ND random variables and let f1(x),…,fn(x) be a sequence of Borel functions which all are monotone increasing (or all are monotone decreasing), then f1(X1),…,fn(Xn) are still ND random variables.Lemma 1.4 (see [7]).
Let{Xn:n≥1} be a sequence of negatively dependent sequence with 𝔼Xn=0, |Xn|≤bn, a.s. (n=1,2,…), t>0, and t·max1≤i≤nbi≤1, then
(1.5)P(|∑i=1n‍Xi|>ε)≤2exp{-tε+t2∑i=1n‍𝔼Xi2},foranyε>0.
In particular, writing σn2=n-1∑i=1n‍var(Xi), t=ε/(2σn2+bnε), we get
(1.6)P(1n|∑i=1n‍Xi|>ε)≤2exp{-nε22(2σn2+bnε)},foranyε>0.


## 2. Main Results

Let{Xn:n≥1} be a sequence of negatively dependent random variables with common distribution function F(x) and density f(x) and let (X(1),…,X(n)) be the order statistic of (X1,…,Xn).In the case of independent observations, in order to estimatef(x) from a sample (X1,…,Xn) Yu [1] considered (1.1) and showed both the pointwise consistency and uniform consistency for fn(x). However, in many situations we are dealing with dependent variables, so we will extend these results for the rate of consistency of nearest neighbor density estimator.Our theorems are formulated in a more general setting.Theorem 2.1.
Let{Xn:n≥1} be a negatively dependent sequence with common distribution function F(x), and 0<F(x)<1. There exists a sequence of positive numbers {τn:n≥1}, {τn:n≥1} which verifies jointly with {mn:n≥1} that
(2.1)mn→∞,τn→0,mnnτn→0,asn→∞.
If f satisfies the local Lipschitz condition at x, and
(2.2)τn2mn2n→∞,
then
(2.3)limn→∞P(|fn(x)-f(x)|>ετn)=0.Theorem 2.2.
Under the conditions of Theorem2.1, If (2.2) instead becomes
(2.4)τn2mn2nlnn→∞,
then
(2.5)|fn(x)-f(x)|=o(τn),a.s.Proof of Theorem2.1.
For anyε>0, we have
(2.6)P(|fn(x)-f(x)|>ετn)≤P(x<X(mn))+P(x≥X(n-mn+1))+P(X(mn)≤x<X(n-mn+1),|fn(x)-f(x)|>ετn)=^Jn1+Jn2+Jn3.
Let's begin with proving ∑n=1∞Jn3<∞. Denote An={X(mn)≤x<X(n-mn+1)}. When An is true, (1.1) implies that there exists j(x), 0≤j(x)≤n-mn, such that x∈[X(mn+j(x)),X(mn+j(x)+1)). Denote X1(x)=X(j(x)+1) and X2(x)=X(2mn+j(x)). Then, we get
(2.7)Jn3≤P(An,fn(x)<f(x)-ετn)+P(An,fn(x)>f(x)+ετn)≤P(An,fn(x)<f(x)-ετn2)+P(An,fn(x)>f(x)+ετn)=^Pn1+Pn2.
Iff(x)≤ετn, then (An,fn(x)<f(x)-ετn) is the impossible event, namely, P(An,fn(x)<f(x)-ετn)=0. Herein below, we suppose that f(x)>ετn for Pn1. Using the previous denotions, we obtain that
(2.8)Pn1=P(X2(x)-X1(x)>2mnn(f(x)-ετn/2),An)≤P(X2(x)-x>mnn(f(x)-ετn/2),An)+P(x-X1(x)>mnn(f(x)-ετn/2),An)=^Pn1′+Pn1′′,Pn2=P(X2(x)-X1(x)<2mnn(f(x)+ετn),An)≤P(X2(x)-x<mnn(f(x)+ετn),An)+P(x-X1(x)<mnn(f(x)+ετn),An)=^Pn2′+Pn2′′.
Write
(2.9)rn=mnn(f(x)-ετn/2),rn′=mnn(f(x)+ετn).
Sincef satisfies the local Lipschitz condition of order one at x, there exist constants M>0 and δ>0, such that |f(t)-f(x)|≤M|t-x| whenever |t-x|<δ, that is,
(2.10)f(x)-M|t-x|≤f(t)≤f(x)+M|t-x|.
From (2.1), mn/nτn→0 and τn→0. Thus, when n is sufficiently large, we have
(2.11)rn=mnn(f(x)-ετn/2)≤mnn(f(x)-f(x)/2)=mn(f(x)/2)nτnτn<ετn4M<δ,rn′=mnn(f(x)+ετn)≤mnnf(x)=mnnτnτnf(x)<ετn2M<δ.
By (2.10), (2.11), we have
(2.12)𝔼I(x≤Xi<x+rn)=∫xx+rn‍f(t)dt≥(f(x)-Mrn)rn=f(x)-Mrnf(x)-ετn/2·mnn≥f(x)-ετn/4f(x)-ετn/2·mnn=4f(x)-ετn4f(x)-2ετn·mnn,(2.13)𝔼I(x≤Xi<x+rn′)=∫xx+rn′‍f(t)dt≤(f(x)+Mrn′)rn′=f(x)+Mrn′f(x)+ετn·mnn≤f(x)+ετn/2f(x)+ετn·mnn=2f(x)+ετn2f(x)+2ετn·mnn,(2.14)𝔼I(x-rn≤Xi<x)=∫x-rnx‍f(t)dt≥(f(x)-Mrn)rn≥4f(x)-ετn4f(x)-2ετn·mnn,(2.15)𝔼I(x-rn′≤Xi<x)=∫x-rn′x‍f(t)dt≤(f(x)+Mrn′)rn′≤2f(x)+ετn2f(x)+2ετn·mnn.
So whenn is sufficiently large, we have by (2.12) and (2.13)
(2.16)Pn1′=P(X2(x)-x>mnn(f(x)-ετn/2),An)≤P(X2(x)-x>rn)=P(∑i=1n‍I(x≤Xi<x+rn)≤mn)≤P(∑i=1n‍(I(x≤Xi<x+rn)-𝔼I(x≤Xi<x+rn))≤mn-n4f(x)-ετn4f(x)-2ετn·mnn)=P(∑i=1n‍ξi≤mn-ετn4f(x)-2ετn<0)≤P(|∑i=1n‍ξi|≥mnετn4f(x)-2ετn)≤P(|∑i=1n‍ξi|≥mnετn4f(x)),
where ξi=I(x≤Xi<x+rn)-𝔼I(x≤Xi<x+rn).
Consider(2.17)Pn2′=P(X2(x)-x<mnn(f(x)+ετn),An)≤P(X2(x)-x<rn′)=P(∑i=1n‍I(x≤Xi<x+rn′)≥mn)≤P(∑i=1n‍(I(x≤Xi<x+rn′)-𝔼I(x≤Xi<x+rn′))≥mn-n2f(x)+ετn2f(x)+2ετn·mnn)=P(∑i=1n‍ξi′≥mnετn2f(x)+2ετn)≤P(∑i=1n‍ξi′≥mnετn4f(x))≤P(|∑i=1n‍ξi′|≥mnετn4f(x)),
where ξi′=I(x≤Xi<x+rn′)-𝔼I(x≤Xi<x+rn′). Likewise, when n is sufficiently large, we have by (2.14) and (2.15)
(2.18)Pn1′′≤P(|∑i=1n‍ηi|≥mnετn4f(x)),(2.19)Pn2′′≤P(|∑i=1n‍ηi′|≥mnετn4f(x)),
where ηi=I(x-rn≤Xi<x)-𝔼I(x-rn≤Xi<x), ηi′=I(x-rn′≤Xi<x)-𝔼I(x-rn′≤Xi<x).
From (2.16) and (2.18), we get
(2.20)Pn1≤P(|∑i=1n‍ξi|≥mnετn4f(x))+P(|∑i=1n‍ηi|≥mnετn4f(x)).
From (2.17) and (2.19), we get
(2.21)Pn2≤P(|∑i=1n‍ξi′|≥mnετn4f(x))+P(|∑i=1n‍ηi′|≥mnετn4f(x)).
Write ζi=I(-∞<Xi<x+rn)-𝔼I(-∞<Xi<x+rn), ζi′=I(-∞<Xi<x+rn′)-𝔼I(-∞<Xi<x+rn′), βi=I(-∞<Xi<x)-𝔼I(-∞<Xi<x).
Then(2.22)P(|∑i=1n‍ξi|≥mnετn4f(x))=P(|∑i=1n‍ζi-∑i=1n‍βi|≥mnετn4f(x))≤P(|∑i=1n‍ζi|≥mnετn8f(x))+P(|∑i=1n‍βi|≥mnετn8f(x))=P(1n|∑i=1n‍ζi|≥amnτnn)+P(1n|∑i=1n‍βi|≥amnτnn),(2.23)P(|∑i=1n‍ξi′|≥mnετn4f(x))=P(|∑i=1n‍ζi′-∑i=1n‍βi|≥mnετn4f(x))≤P(|∑i=1n‍ζi′|≥mnετn8f(x))+P(|∑i=1n‍βi|≥mnετn8f(x))=P(1n|∑i=1n‍ζi′|≥amnτnn)+P(1n|∑i=1n‍βi|≥amnτnn),
where a=ε/8f(x).
Obviously,{ζi}, {ζi′}, and {βi} are monotonic function’s. According to Lemma 1.3, {ζi}, {ζi′}, and {βi} are still ND sequences. We will apply Lemma 1.4 to {ζi}, {ζi′}, and {βi}. For {ζi} and {ζi′} we have 𝔼(ζi)=0,𝔼(ζi′)=0, |ζi|=|I(-∞<Xi<x+rn)-𝔼I(-∞<Xi<x+rn)|≤|I(-∞<Xi<x+rn)|+|𝔼I(-∞<Xi<x+rn)|≤2, |ζi′|≤2, n-1∑i=1n‍var(ζi)=var(I(-∞<Xi<x+rn)-𝔼I(-∞<Xi<x+rn))≤1, n-1∑i=1n‍var(ζi′)≤1. By (2.1), when n is sufficiently large, such that τnmn/n<1, then we have
(2.24)P(1n|∑i=1n‍ζi|≥amnτnn)≤2exp{-n(amnτn/n)22(2+2(amnτn/n))}=2exp{-cmn2τn2n},(2.25)P(1n|∑i=1n‍ζi′|≥amnτnn)≤2exp{-cmn2τn2n},
where c=a2/4(1+a).
By (2.2) and (2.24), then we have
(2.26)P(1n|∑i=1n‍ζi|≥amnτnn)→0.
By (2.2) and (2.25), then we have
(2.27)P(1n|∑i=1n‍ζi′|≥amnτnn)→0.
Similarly,
(2.28)P(1n|∑i=1n‍βi|≥amnτnn)→0.
From (2.22), (2.26), and (2.28), we get P(|∑i=1n‍ξi|≥mn(ετn/4f(x)))→0. Similarly, P(|∑i=1n‍ηi|≥mn(ετn/4f(x)))→0. By (2.20), then Pn1→0. From (2.23), (2.27), and (2.28), we get P(|∑i=1n‍ξi′|≥mn(ετn)/4f(x))→0. Similarly, P(|∑i=1n‍ηi′|≥mn(ετn/4f(x)))→0. By (2.21), then Pn2→0. Hereby, we have proved that Jn3→0. It remains to show that Jn1→0 and Jn2→0. Denote
(2.29)βi′=I(x<Xi<∞)-𝔼I(x<Xi<∞).
According to the conditions of Theorem2.1, we can see 0<𝔼I(-∞<Xi<x)=P(Xi<x)=F(x)<1, mn/n→0,asn→∞. Then, there exist positive constants a1 and a2 such that 𝔼I(-∞<Xi<x)-mn/n>a1 and 𝔼I(x<Xi<∞)-mn/n>a2. When n is sufficiently large, we have
(2.30)Jn1=P(x<X(mn))=P(∑i=1n‍I(-∞<Xi<x)≤mn)=P(∑i=1n‍βi≤mn-n𝔼I(-∞<Xi<x)<-a1n)≤P(|1n∑i=1n‍βi|>a1),Jn2=P(x≥X(n-mn+1))=P(∑i=1n‍I(x<Xi<∞)≤mn)=P(∑i=1n‍βi′≤mn-n𝔼I(x<Xi<∞)<-a2n)≤P(|1n∑i=1n‍βi′|>a2).
We will apply Lemma1.4 to {βi} and {βi′}. For {βi}, we have 𝔼(βi)=0,|βi|=|I(-∞<Xi<x)-𝔼I(-∞<Xi<x)|≤|I(-∞<Xi<x)|+|𝔼I(-∞<Xi<x)|≤2, σn2=n-1∑i=1n‍var(βi)=var(βi)=var(I(-∞<Xi<x)-𝔼I(-∞<Xi<x))≤1. When n is sufficiently large, then we have
(2.31)P(|1n∑i=1n‍βi|>a1)≤2exp{-na122(2σn2+2a1)}≤2exp{-na122(2+2a1)}=2exp{-c′n}→0,
where c′=a12/2(2+2a1). Similarly,
(2.32)P(|1n∑i=1n‍βi′|>a2)≤2exp{-na222(2+2a2)}→0.
From (2.31) and (2.32), we can get Jn1→0,Jn2→0. This completes the proof of Theorem 2.1.Proof of Theorem2.2.
By (2.4), when n is sufficiently large, we get cτn2mn2/n>2lnn=lnn2. From (2.24), then we have
(2.33)P(1n|∑i=1n‍ζi|≥amnτnn)≤2exp{-cτn2mn2n}≤2exp(-lnn2)=2n2.
Similarly,
(2.34)P(1n|∑i=1n‍ηi|≥amnτnn)≤2n2.
From (2.33) and (2.34), we get ∑n=1∞Pn1<∞. For Pn2, the proof follows in the same way. From (2.7), we can get ∑n=1∞Jn3<∞. Similarly, by (2.30), we have ∑n=1∞Jn1<∞ and ∑n=1∞Jn2<∞. From (2.6), we can get
(2.35)∑n=1∞‍P(|fn(x)-f(x)|>ετn)<∞.
According to the Borel-Cantelli lemma, we have(2.36)|fn(x)-f(x)|=o(τn).a.s.



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*Source: 569547-2012-11-25.xml*