- Research
- Open Access
The large deviation for the least squares estimator of nonlinear regression model based on WOD errors
- Xufeng Huang^{1},
- Xufei Tang^{2},
- Xin Deng^{2} and
- Xuejun Wang^{2}Email author
https://doi.org/10.1186/s13660-016-1064-6
© Huang et al. 2016
- Received: 8 October 2015
- Accepted: 28 March 2016
- Published: 23 April 2016
Abstract
As a kind of dependent random variables, the widely orthant dependent random variables, or WOD for short, have a very important place in dependence structures for the intricate properties. And so its behavior and properties in different statistical models will be a major part in our research interest. Based on WOD errors, the large deviation results of the least squares estimator in the nonlinear regression model are established, which extend the corresponding ones for independent errors and some dependent errors.
Keywords
- widely orthant dependent random variables
- large deviation
- nonparametric regression model
- least squares estimator
MSC
- 62J02
- 62F12
1 Introduction
Many researchers have paid attention to the study of the probability limit theorem and its applications for the independent random variables, while the fact is that most of the random variables found in real practice are dependent, which just motivates the authors’ interests in how well the dependent random variables will behave in some cases.
One of the important dependence structures is the widely orthant dependence structure. The main purpose of the paper is to study the large deviation for the least squares estimator of the nonlinear regression model based on widely orthant dependent errors.
1.1 Brief review
Noting that \(Q(x_{1},x_{2},\ldots,x_{n};\theta)=Q_{n}(\theta)\) is defined on \(\mathbf{R}^{n}\times\Theta\), where Θ is compact. Furthermore, \(Q (x;\theta)\), where \(x=(x_{1},x_{2},\ldots,x_{n})\), is a Borel measurable function of x for any fixed \(\theta\in\Theta\) and a continuous function of θ for any fixed \(x\in\mathbf{R}^{n}\). Lemma 3.3 of Schmetterer [1] shows that there exists a Borel measurable map \(\theta_{n}:\mathbf{R}^{n}\to\Theta\) such that \(Q_{n}(\theta_{n})=\inf_{\theta\in\Theta}Q_{n}(\theta)\). In the following, we will consider this version as the least squares estimator \(\theta_{n}\).
Let \(\theta_{0}\) be the true parameter and assume that \(\theta_{0}\in\Theta\). Ivanov [2] established the following large deviation result for independent and identically distributed (i.i.d.) random variables.
Theorem 1.1
Inspired by the above literature, we will establish the large deviation results based on widely orthant dependent errors.
1.2 Concept of widely orthant dependence structure
In this section, we will present the widely orthant dependence structure, which was introduced by Wang et al. [9].
Definition 1.1
An array \(\{X_{ni},i\geq1,n\geq1\}\) of random variables is called a row-wise WOD if for every \(n\geq1\), \(\{X_{ni},i\geq1\}\) is a sequence of WOD random variables.
As mentioned above, Wang et al. [9] first introduced the concept of WOD random variables. Their properties and applications have been studied consequently. For instance, WOD random variables include some common negatively dependent random variables, some positively dependent random variables and others, which were shown in the examples provided by Wang et al. [9] and the uniform asymptotic for the finite-time ruin probability of a new dependent risk model with a constant interest rate was also investigated in the same work. He et al. [10] established the asymptotic lower bounds of precise large deviations with non-negative and dependent random variables. The uniform asymptotic for the finite time ruin probabilities of two types of non-standard bidimensional renewal risk models with constant interest forces and diffusion generated by Brownian motions was proposed by Chen et al. [11]. The Bernstein type inequality for WOD random variables and its applications were studied by Shen [12]. Wang et al. [13] investigated the complete convergence for WOD random variables and gave its applications to nonparametrics regression models, and so forth.
As is well known, the class of WOD random variables contains END random variables, NOD random variables, NSD random variables, NA random variables, and independent random variables as special cases. Hence, it is meaningful to extend the results of Yang and Hu [5] to WOD errors.
Throughout this paper, let \(\{\xi_{i}, i \geq1\}\) be a sequence of WOD random variables with dominating coefficients \(f_{U}(n)\), \(f_{L}(n)\), \(n \geq1\). Denote \(f(n)=\max \{f_{U}(n),f_{L}(n) \}\). Let C denote a positive constant, which may vary in different spaces. Let \(\lfloor x \rfloor\) be the integer part of x.
The main results and their proofs are presented in Section 3 and for the convenience of the reader, some useful lemmas relating to the proofs are listed in Section 2.
2 Preliminary lemmas
In this section, we provide some important lemmas will be used to prove the main results of the paper. The first one is the basic property for WOD random variables, which was established by Wang et al. [13].
Lemma 2.1
- (i)
If \(\{h_{n}(\cdot), n \geq1\}\) are all non-decreasing (or all non-increasing), then \(\{h_{n}(X_{n}), n \geq1\}\) are still WOD.
- (ii)For each \(n\geq1\) and any \(s\in\mathbf{R}\),$$ E\exp \Biggl\{ s\sum^{n}_{i=1}X_{i} \Biggr\} \leq f(n)\prod^{n}_{i=1}E\exp \{sX_{i}\}. $$(2.1)
The next lemma is very useful to prove the main results of the paper, which can be found in Hu [4].
Lemma 2.2
The following are the Marcinkiewicz-Zygmund type inequality and Rosential-type inequality for WOD random variables, which play an important role in the proof.
Lemma 2.3
(cf. Wang et al. [13])
3 Main results and their proofs
Based on the useful inequalities in Section 2, we now study the large deviation results for the least squares estimator of the nonlinear regression model based on WOD errors.
Theorem 3.1
Proof
Inspired by Theorem 3.1, we will consider the case \(p\in(1,2]\) and establish the following result.
Theorem 3.2
Proof
Declarations
Acknowledgements
This work was supported by the National Natural Science Foundation of China (11501004, 11501005, 11526033), the Natural Science Foundation of Anhui Province (1508085J06), the Students Science Research Training Program of Anhui University (KYXL2014017), and the Research Teaching Model Curriculum of Anhui University (xjyjkc1407).
The authors are most grateful to the editor and anonymous referees for careful reading of the manuscript and valuable suggestions, which helped in improving an earlier version of this paper.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Schmetterer, L: Introduction to Mathematical Statistics. Springer, Berlin (1974) View ArticleMATHGoogle Scholar
- Ivanov, AV: An asymptotic expansion for the distribution of the least squares estimator of the nonlinear regression parameter. Theory Probab. Appl. 21(3), 557-570 (1976) View ArticleMATHGoogle Scholar
- Hu, SH: The rate of convergence for the least squares estimator in nonlinear regression model with dependent errors. Sci. China Ser. A 45(2), 137-146 (2002) MathSciNetMATHGoogle Scholar
- Hu, SH: Consistency for the least squares estimator in nonlinear regression model. Stat. Probab. Lett. 67(2), 183-192 (2004) MathSciNetView ArticleMATHGoogle Scholar
- Yang, WZ, Hu, SH: Large deviation for a least squares estimator in a nonlinear regression model. Stat. Probab. Lett. 91, 135-144 (2014) MathSciNetView ArticleMATHGoogle Scholar
- Ibragimov, IA, Has’minskii, RZ: Statistical Estimation: Asymptotic Theory. Springer, New York (1981). Translated by Samuel Kotz View ArticleMATHGoogle Scholar
- Ivanov, AV, Leonenko, NN: Statistical Analysis of Random Fields. Kluwer Academic Publishers, Dordreht/Boston/London (1989) View ArticleMATHGoogle Scholar
- Ivanov, AV: Asymptotic Theory of Nonlinear Regression. Kluwer Academic Publishers, Dordreht/Boston/London (1997) View ArticleMATHGoogle Scholar
- Wang, K, Wang, Y, Gao, Q: Uniform asymptotics for the finite-time ruin probability of a new dependent risk model with a constant interest rate. Methodol. Comput. Appl. Probab. 15(1), 109-124 (2013) MathSciNetView ArticleMATHGoogle Scholar
- He, W, Cheng, DY, Wang, YB: Asymptotic lower bounds of precise large deviations with nonnegative and dependent random variables. Stat. Probab. Lett. 83, 331-338 (2013) MathSciNetView ArticleMATHGoogle Scholar
- Chen, Y, Wang, L, Wang, YB: Uniform asymptotics for the finite-time ruin probabilities of two kinds of nonstandard bidimensional risk models. J. Math. Anal. Appl. 401(1), 114-129 (2013) MathSciNetView ArticleMATHGoogle Scholar
- Shen, AT: Bernstein-type inequality for widely dependent sequence and its application to nonparametric regression models. Abstr. Appl. Anal. 2013, Article ID 862602 (2013) MathSciNetGoogle Scholar
- Wang, XJ, Xu, C, Hu, TC, Volodin, A, Hu, SH: On complete convergence for widely orthant-dependent random variables and its applications in nonparametrics regression models. Test 23(3), 607-629 (2014) MathSciNetView ArticleMATHGoogle Scholar