Asymptotics of M-estimator in multivariate linear regression models for a class of random errors

IF 0.8 4区数学 Q3 STATISTICS & PROBABILITY

Australian & New Zealand Journal of Statistics Pub Date : 2023-07-21 DOI:10.1111/anzs.12393

Yi Wu, Wei Yu, Xuejun Wang

引用次数: 0

Abstract

It is known that linear regression models have immense applications in various areas such as engineering technology, economics and social sciences. In this paper, we investigate the asymptotic properties of M-estimator in multivariate linear regression model based on a class of random errors satisfying a generalised Bernstein-type inequality. By using the generalised Bernstein-type inequality, we obtain a general result on almost sure convergence for a class of random variables and then obtain the strong consistency for the M-estimator in multivariate linear regression models under some mild conditions. The result extends or improves some existing ones in the literature. Moreover, we also consider the case when the dimension $p$ tends to infinity by establishing the rate of almost sure convergence for a class of random variables satisfying generalised Bernstein-type inequality. Some numerical simulations are also provided to verify the validity of the theoretical results.

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一类随机误差的多元线性回归模型中M-估计的渐近性

众所周知，线性回归模型在工程技术、经济和社会科学等各个领域都有着巨大的应用。本文研究了基于一类满足广义Bernstein型不等式的随机误差的多元线性回归模型中M-估计量的渐近性质。利用广义Bernstein型不等式，我们得到了一类随机变量几乎肯定收敛的一般结果，并在一些温和条件下得到了多元线性回归模型中M-估计量的强一致性。该结果扩展或改进了文献中已有的一些结果。此外，我们还通过建立一类满足广义Bernstein型不等式的随机变量的几乎肯定收敛率，来考虑维数$p$趋于无穷大的情况。通过数值模拟验证了理论结果的正确性。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Australian & New Zealand Journal of Statistics 数学-统计学与概率论

CiteScore

1.30

自引率

9.10%

发文量

审稿时长

>12 weeks

期刊介绍： The Australian & New Zealand Journal of Statistics is an international journal managed jointly by the Statistical Society of Australia and the New Zealand Statistical Association. Its purpose is to report significant and novel contributions in statistics, ranging across articles on statistical theory, methodology, applications and computing. The journal has a particular focus on statistical techniques that can be readily applied to real-world problems, and on application papers with an Australasian emphasis. Outstanding articles submitted to the journal may be selected as Discussion Papers, to be read at a meeting of either the Statistical Society of Australia or the New Zealand Statistical Association. The main body of the journal is divided into three sections. The Theory and Methods Section publishes papers containing original contributions to the theory and methodology of statistics, econometrics and probability, and seeks papers motivated by a real problem and which demonstrate the proposed theory or methodology in that situation. There is a strong preference for papers motivated by, and illustrated with, real data. The Applications Section publishes papers demonstrating applications of statistical techniques to problems faced by users of statistics in the sciences, government and industry. A particular focus is the application of newly developed statistical methodology to real data and the demonstration of better use of established statistical methodology in an area of application. It seeks to aid teachers of statistics by placing statistical methods in context. The Statistical Computing Section publishes papers containing new algorithms, code snippets, or software descriptions (for open source software only) which enhance the field through the application of computing. Preference is given to papers featuring publically available code and/or data, and to those motivated by statistical methods for practical problems.