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Constrained M-estimation for linear regression models

机译：线性回归模型的约束M估计

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摘要

In this thesis we propose a new class of estimates for the linear regression model: The Constrained M-estimates of regression and scale, or CM-estimates of regression for short. The CM-estimates of regression are affine and regression equivariant and have breakdown point equal to 0.5. In addition, these estimates can be tuned to possess good local robustness properties.;In Chapter 2 we introduce the CM-estimates of regression and scale. Several theoretical results are obtained. After introducing the concept of CM-functionals of regression and scale, we prove existence and uniqueness of the CM-functionals and consistency of the CM-estimates. Then we derive the expression for their finite sample replacement breakdown point, and obtain the influence function of the CM-functional of regression and the asymptotic normal distribution of the regression CM-estimates. We derive indices of local robustness based on the influence function of the CM-functional of regression and based on the asymptotic covariance matrix of the regression CM-estimates, and propose a procedure for locally tuning the CM-estimates. We illustrate this procedure by finding the tuning constant for the Tukey's biweighted class of CM-estimates, assuming the errors follow a normal distribution. We show how we can improve upon the relative asymptotic efficiency and the residual gross error sensitivity of the biweighted S-estimates.;In Chapter 3 we study the special case (p = 1 and intercept) of the CM-estimates of regression: The Constrained M-estimates of univariate location and scale. We find the tuning constant for two classes of location CM-estimates, namely the Tukey's biweighted and the inverse exponential, and illustrate their good global properties by computing the estimates for two well-known contaminated data sets.;In Chapter 4 we investigate the bias behavior of the CM-estimates studied in Chapters 2 and 3. By performing a Monte Carlo study, we investigate the stability of a selection of location estimators in small and large samples. For these simulations we assumed several situations where the underlying distribution was subject to different types of point mass contamination under different percentages of contamination. In the regression setting we obtain an approximation for the maximum asymptotic bias of the regression CM-functionals by performing large samples simulations, and show that the properly tuned regression CM-estimates compare favorably to other high breakdown point estimates.
机译：在本文中，我们为线性回归模型提出了一类新的估计：回归和规模的约束M估计，或简称CM估计。回归的CM估计是仿射和回归等变的，并且崩溃点等于0.5。此外，这些估计可以调整为具有良好的局部鲁棒性。在第二章中，我们介绍了回归和规模的CM估计。获得了一些理论结果。在介绍了回归和尺度的CM函数的概念之后，我们证明了CM函数的存在和唯一性以及CM估计的一致性。然后，导出其有限样本替换分解点的表达式，并获得回归的CM函数的影响函数和回归CM估计的渐近正态分布。我们基于回归的CM函数的影响函数和回归CM估计的渐近协方差矩阵，得出局部鲁棒性指标，并提出了局部调整CM估计的程序。我们假设误差遵循正态分布，通过找到Tukey的CM估计的双加权类的调整常数来说明此过程。我们展示了如何改善双加权S估计的相对渐近效率和剩余总误差敏感性。;在第3章中，我们研究了回归的CM估计的特殊情况（p = 1和截距）：单变量位置和规模的M估计。我们找到了两类位置CM估计的调谐常数，即Tukey的双加权和反指数，并通过计算两个著名的污染数据集的估计来说明它们的良好全局特性。在第4章中，我们研究了偏差。第2章和第3章研究的CM估计的行为。通过执行蒙特卡洛研究，我们调查了大小样本中选择的位置估计量的稳定性。对于这些模拟，我们假设了几种情况，其中基础分布在不同百分比的污染下会受到不同类型的点质量污染。在回归设置中，我们通过执行大型样本模拟获得回归CM功能的最大渐近偏差的近似值，并显示正确调整的回归CM估计与其他高分解点估计相比具有优势。

著录项

  • 作者

    Mendes, Beatriz Vaz de Melo.;

  • 作者单位

    Rutgers The State University of New Jersey - New Brunswick.;

  • 授予单位 Rutgers The State University of New Jersey - New Brunswick.;
  • 学科 Statistics.
  • 学位 Ph.D.
  • 年度 1995
  • 页码 125 p.
  • 总页数 125
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类
  • 关键词

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