The consistency of estimator under fixed design regression model with NQD errors

In this article, basing on NQD samples, we investigate the fixed design nonparametric regression model, where the errors are pairwise NQD random errors, with fixed design points, and an unknown function. Nonparametric weighted estimator will be introduced and its consistency is studied. As special case, the consistency result for weighted kernel estimators of the model is obtained. This extends the earlier work on independent random and dependent random errors to NQD case.


Introduction
In regression analysis, it is common practice to investigate the functional relationship between the responses and design points. Nonparametric regression model provides a useful explanatory and diagnostic tool for this purpose. One may see Muller [1] and Hardle [2] for many examples about this and good introductions to the general subject area.
To begin with, consider the fixed design nonparametric regression model in the paper Y nk = g(x nk ) + ε nk , 1 ≤ k ≤ n.
Here x nk , 1 ≤ k ≤ n , are known fixed design points, and ε nk are random errors , g(·) is an unknown regression function. As an estimate of g(·), we consider the following general linear smoother.
It is well known that Georgiev [3] first proposed the estimator above, and the estimator subsequently have been studied by many authors. A brief review of the theoretic development in recent years is worth mentioning. Results on ε nk being assumed to be independent, consistency and asymptotic normality have been investigated by Georgiev [4] and Müller [5] among others. Results for the case when ε nk are dependent have also been studied by various authors in recent years. Roussas et al. [6] established asymptotic normality of g n (x) assuming that the errors are from a strictly stationary stochastic process under the strong mixing condition. Tran et al. [7] discussed again asymptotic normality of g n (x) assuming that the errors form a weakly stationary linear process with a martingale difference sequence.
Hu et al. [8] gave the mean consistency, complete consistency, and asymptotic normality of regression models based on linear process errors. Under negatively associated sequences, Liang and Jing [9] presented some asymptotic properties for estimates of nonparametric regression models, Yang et al. [10] generalized part results of Liang and Jing [9] for negatively associated sequences to the case of negatively orthant dependent sequences, and so on.
In this paper, we shall investigate the above nonparametric regression problem under pairwise NQD errors, which means more general case for sampling. Definition 1.1. [11] The pair (X, Y ) of random variables X and Y is said to be NQD (1.1) A sequence of random variables {X n , n ≥ 1} is pairwise NQD random variables(for short, It can be deduced from Definition 1.1 that Moreover, it follows that (1.2) also implies (1.1), and hence, (1.1) and (1.2) are actually equivalent.
The definition was introduced by Lehmann [11], which contains independent random variable, NA(negatively associated) random variable and NOD(negatively orthant dependent) random variable et al. as special cases. For the reason of the wide applications of NQD random variables in reliability theory and applications, the notions of NQD random variables have received many concern recently. Some properties about NQD random variables can be found in Lehmann [4], and there are many other meaningful literature (e.g. Matula [12], Huang et al. [13], Sung [14], Shi [15], Wang et al. [17], Li et al. [18]).
However, the pairwise NQD structure is more comprehensive than the NA (negative associated) structure and the NOD (negatively orthant dependent) structure. Concerning to the study for the theory of pairwise NQD random variables, due to lack of some key technique tool, such as Bernstain type inequality and exponential inequality etc. still unestablished for NQD sequences, investigating related result is restraint, especially the estimators of parametric and nonparametric components in regressions model under NQD error's structure.
Hence, extending the asymptotic properties of independent and other dependent random variables to the case of NQD variables is highly desirable and of considerably significance in the theory and application.
In this article, basing on several related lemmas, we investigate the fixed design nonparametric regression model with NQD errors. Nonparametric estimator g n (·) of g(·) will be introduced and its usual consistency properties of g n (·) including mean convergence, uniform mean convergence, convergence in probability, et al. are studied under suitable regularity conditions.
The organization of this paper is as follows. In section 2, we shall present several lemmas for proof of main results, and give the basic assumptions for the nonparametric estimator.
We give the further assumption and the main results in section 3. The proofs of the results will be deferred to Section 4.
2 Some lemmas and Basic assumptions

Some lemmas
We shall begin with a few preliminary lemmas useful in the proofs of our main results.
Firstly, a fact about the NQD properties is cited from [11].
Lemma 2.1. [11] Let the pair (X, Y ) of random variables X and Y be NQD, then for any x, y ∈ R; (3) If f, g are both non-decreasing (or non-increasing) functions, then f (X) and g(X) are NQD.
Then, according to Condition (A 2 ), we conclude Note that by the definition of H(·), lim for all x ∈ (0, 1) and Theorem, which together with (2.3) implies (2.1).
Again because of ∞ −∞ |K(u)| du < ∞ and h n → 0, if n large sufficiently, one can choose a sufficient small positive number τ 0 , such that when |h n u| < τ 0 < τ , there is As a result, for x ∈ [τ, 1 − τ ], uniformly Combining (2.3), then (2.2) holds, as we wanted to show. This completes the proof.

4)
and for a fixed point τ ∈ (0, 1/2), Proof of Lemma 2.4 The proof is similar to those of Lemma 2.3 with |K(u)| replaced by K(u) and using Condition (A 3 ), so is omitted here.

Basic assumptions
Unless otherwise specified, we assume throughout the paper that the random sample (x nk , Y nk ) for 1 ≤ k ≤ n come from the regression model where {ε nk , 1 ≤ k ≤ n} from a sequence of zero mean random errors with the same distribution as {ε k , 1 ≤ k ≤ n} for each n, {x nk , 1 ≤ k ≤ n} are known fixed design points from a compact set A in R d (d is a positive integer), and g(·) is an unknown real valued regression function and assumed to be bounded on the compact set A.
The present paper investigates the general linear smoother as an estimate of g(·) in the following, defined by formula where the array of weight functions ω nk (x), 1 ≤ k ≤ n depends on the fixed design points x, x n1 , · · · , x nn and on the number of observations n, which ω nk (x) = 0 for k > n.
In the following section, we denote all continuity points of the function g(·) on set A as C(g). Let the symbol x be the Eucledean norm of x , M a generic positive constant in the sequel, which could take different values at different places.

Main results
We shall establish two different models of convergence for the nonparametric regression The weights ω nk (x), 1 ≤ k ≤ n in the assumptions is relatively extensive in practice, which can be easily satisfied by the commonly adopted weights used, such as the well-known nearest neighbor weights.
When |x ni − x| = |x nj − x|, assume that |x ni − x| is ahead of |x nj − x| for x ni < x nj , then a permutation for |x n1 − x| , |x n2 − x| , · · · , |x nn − x| can be given as follows Let k n = o(n), if define the nearest neighbor weight as otherwise.
Then, one can easily verify by the choice of x ni and the definition of R i (x) that Conditions We now state our first result for the mean convergence of g n (x), which, on the opinion of statistics, is asymptotically unbiased of g(x) in the proof of Theorem 3.1.
Another similar form of mean convergence, by using the inequality ( n k=1 |a  for 0 < p ≤ 2. Remark 3.1. Since NA sequence and NOD sequence are NQD sequence, we generalize some results of Liang et al. [9]and Yang et al. [10] to the case of NQD errors, respectively. And as a consequence, one may get consistency property for the weighted kernel estimators in the model (2.6).
Next, we shall give the weak consistency for the estimator of g(x) under existence of absolute mean for variable. for ∀x ∈ C(g). x n(k) − x n(k−1) , in probability as n → ∞, for ∀x ∈ (0, 1).
Proof of Theorem 3.2 Note that in a compact set A, g(·) is uniformly continuous if it is continuous. Consequently, similar proof as Theorem 3.1, we can get that tends to zero if n → ∞, which means the desired result (3.2).
Proof of Corollary 3.1 Note that under Condition (A 2 ), And there are This ends the proof.
Proof of Theorem 3.3 Since x ∈ C(g), the same reason as before, for any ε > 0, there is a number δ > 0, when a ∈ (0, δ), one may get that |Eg n (x) − g(x)| tends to zero by arbitrary of ε > 0 and Conditions (B 1 ), (B 2 ), (B 4 ). For proving (3.5), note that We now prove that the random part of r.h.s. in (4.4) tends to zero in probability as n → ∞ . Observe that Next introduce truncated variables below.
When it come to I n1 , as n → ∞.
Proof of Corollary 3.2 By the discussion in Corollary 3.1, it is the direct result of Theorem 3.3. This completes the proof of Corollary 3.2.

Competing interests
The authors declare that they have no competing interests.