Adaptive Regression

Linear regression is an important area of statistics, theoretical or applied. There have been a large number of estimation methods proposed and developed for linear regression. Each has its own competitive edge but none is good for all purposes. This manuscript focuses on construction of an adaptive combination of two estimation methods. The purpose of such adaptive methods is to help users make an objective choice and to combine desirable properties of two estimators.

This book presents some recent developments in the theory of robust estimation of linear regression models. It will serve as a reference book in post graduate courses.

1 Prologue.- 1.1 Introduction.- 1.2 Adaptive Combination of Estimators.- 1.3 Notes.- 2 Regression Methods.- 2.1 Introduction.- 2.2 LS Regression.- 2.3 Ridge Regression.- 2.4 LAD Regression.- 2.5 M-Regression.- 2.6 L-Regression.- 2.7 Other Regression Estimators.- 2.8 Estimators of Scale Parameter.- 2.9 Notes.- 3 Adaptive LAD + LS Regression.- 3.1 Introduction.- 3.2 Convex Combination of LAD and LS Regressions.- 3.3 Adaptive Combination of LAD and LS Regressions.- 3.4 Illustrative Examples.- 3.5 Notes.- 4 Adaptive LAD + TLS Regression.- 4.1 Introduction.- 4.2 Adaptive Combination of LAD and Trimmed LS.- 4.3 An Example of Multiple Regression.- 4.4 Notes.- 5 Adaptive LAD + M-Regression.- 5.1 Introduction.- 5.2 Combination of LAD and M-Estimators.- 5.3 Adaptive Combination of LAD and M-Estimators.- 5.4 An Example of Multiple Regression.- 5.5 Notes.- 6 Adaptive LS + TLS Regression.- 6.1 Introduction.- 6.2 Adaptive Combination of Mean and Trimmed Mean.- 6.3 Adaptive Combination of LS and TLS Regressions.- 6.4 Example of Multiple Regression.- 6.5 Notes.- 7 Adaptive Choice of Trimming.- 7.1 Introduction.- 7.2 Fully Adaptive Trimmed Mean and TLS.- 7.3 Adaptive Choice for fhe Trimmed Mean.- 7.4 Adaptive Choice in Linear Model Based on Ranks.- 7.5 Adaptive Choice in Linear Model Based on Regression Rank Scores.- 7.6 Notes.- 8 Adaptive Combination of Tests.- 8.1 Introduction.- 8.2 Types of Tests.- 8.3 Adaptive Combination of F-Test and Median-Type Test.- 8.4 Adaptive Combination of M-Test and Median-Type Test.- 8.5 Notes.- 9 Computational Aspects.- 9.1 Introduction.- 9.2 Computing the Adaptive Combination of LS and LAD.- 9.3 Program ADAPTIVE.- 10 Some Asymptotic Results.- 10.1 Asymptotic Properties of Studentized M-Estimators.- 10.2 Uniform Asymptotic Linearity of M-Statistics.- 10.3 Estimators of Scale Parameter.- 10.4 Optimal Choice of ?n.- 11 Epilogue.- References.- Author Index.