^{1}

We study the strong consistency of estimator of fixed design regression model under negatively dependent sequences by using the classical Rosenthal-type inequality and the truncated method. As an application, the strong consistency for the nearest neighbor estimator is obtained.

Let

Consider the following fixed design regression model:

The above estimator was first proposed by Georgiev [

The concept of negatively dependent random variables was introduced by Lehmann [

A finite collection of random variables

Obviously, independent random variables are ND. Joag-Dev and Proschan [

Let

So we can see that ND is weaker than NA. A number of well-known multivariate distributions have the ND properties, such as multinomial, convolution of unlike multinomials, multivariate hypergeometric, dirichlet, dirichlet compound multinomial, and multinomials having certain covariance matrices. Because of the wide applications of ND random variables, the limiting behaviors of ND random variables have received more and more attention recently. A number of useful results for ND random variables have been established by many authors. We refer to Volodin [

This work is organized as follows: some preliminary lemmas are presented in Section

Throughout the paper,

In this section, we will present some important lemmas which will be used to prove the main results of the paper.

Let the random variables

Let

The following concept of stochastic domination will be used in this work.

A sequence

By the definition of stochastic domination and integration by parts, we can get the following property for stochastic domination. For the details of the proof, one can refer to Wu [

Let

Unless otherwise specified, we assume throughout the paper that

Based on the assumptions above, we can get the following strong consistency of the fixed design regression estimator

Let

For

By (

For fixed

By

For

For

Next, we will prove that

Since

Finally, we will prove that

As an application of Theorem

Let

Based on the notations above, we can get the following result by using Theorem

Let

It suffices to show that the conditions of Theorem

For any

The authors are most grateful to the Editor Michiel Bertsch and anonymous referee for careful reading of the paper and valuable suggestions which helped in improving an earlier version of this paper. This work was supported by the National Natural Science Foundation of China (11201001, 11171001), the Natural Science Foundation of Anhui Province (1208085QA03, 1308085QA03), Applied Teaching Model Curriculum of Anhui University (XJYYXKC04), Doctoral Research Start-up Funds Projects of Anhui University, Students Innovative Training Project of Anhui University (201310357004), and the Students Science Research Training Program of Anhui University (KYXL2012007, kyxl2013003).