Abstract

Let be a sequence of negatively dependent random variables. Based on , 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.

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 is an integer, the nonparametric estimate of is defined by dividing by times the length of the smallest interval containing which consists of of the observations in which half of them lie on the left side of and half on the right side. Formally, where is the order statistic of . In Yu [1] It is showed that in the case of independent and identically distributed observations converges to in probability for almost all and converges to with probability one for almost all . Moreover, the strong uniform consistency of this estimator for 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 -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 variables and are said to be negatively dependent (ND) if for all . 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 for all . 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 variables are said to be negatively dependent (ND) if for all real , An infinite random sequence is said to be ND if every finite subset is ND.

Lemma 1.3 (see [2]). Let be random variables and let be a sequence of Borel functions which all are monotone increasing (or all are monotone decreasing), then are still random variables.

Lemma 1.4 (see [7]). Let be a sequence of negatively dependent sequence with , , a.s. , , and , then In particular, writing , , we get

2. Main Results

Let be a sequence of negatively dependent random variables with common distribution function and density and let be the order statistic of .

In the case of independent observations, in order to estimate from a sample Yu [1] considered (1.1) and showed both the pointwise consistency and uniform consistency for . 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 be a negatively dependent sequence with common distribution function , and . There exists a sequence of positive numbers , which verifies jointly with that If satisfies the local Lipschitz condition at , and then

Theorem 2.2. Under the conditions of Theorem 2.1, If (2.2) instead becomes then

Proof of Theorem 2.1. For any , we have Let's begin with proving . Denote . When is true, (1.1) implies that there exists , , such that . Denote and . Then, we get
If , then is the impossible event, namely, . Herein below, we suppose that for . Using the previous denotions, we obtain that Write
Since satisfies the local Lipschitz condition of order one at , there exist constants and , such that whenever , that is,
From (2.1), and . Thus, when is sufficiently large, we have
By (2.10), (2.11), we have
So when is sufficiently large, we have by (2.12) and (2.13) where .
Consider where . Likewise, when is sufficiently large, we have by (2.14) and (2.15) where , .
From (2.16) and (2.18), we get
From (2.17) and (2.19), we get Write , , .
Then where .
Obviously, , , and are monotonic function’s. According to Lemma 1.3, , , and are still sequences. We will apply Lemma 1.4 to , , and . For and we have , , , , . By (2.1), when is sufficiently large, such that , then we have where .
By (2.2) and (2.24), then we have
By (2.2) and (2.25), then we have Similarly,
From (2.22), (2.26), and (2.28), we get . Similarly, . By (2.20), then . From (2.23), (2.27), and (2.28), we get . Similarly, . By (2.21), then . Hereby, we have proved that . It remains to show that and . Denote
According to the conditions of Theorem 2.1, we can see , . Then, there exist positive constants and such that and . When is sufficiently large, we have
We will apply Lemma 1.4 to and . For , we have , . When is sufficiently large, then we have where . Similarly, From (2.31) and (2.32), we can get . This completes the proof of Theorem 2.1.