Edgeworth expansion and bootstrap approximation for M-estimators of linear regression PDF
Preview Edgeworth expansion and bootstrap approximation for M-estimators of linear regression
Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 1991 Edgeworth expansion and bootstrap approximation for M-estimators of linear regression parameters with increasing dimensions Muhammad Arif Tiro Iowa State University Follow this and additional works at:https://lib.dr.iastate.edu/rtd Part of theStatistics and Probability Commons Recommended Citation Tiro, Muhammad Arif, "Edgeworth expansion and bootstrap approximation for M-estimators of linear regression parameters with increasing dimensions " (1991).Retrospective Theses and Dissertations. 10077. https://lib.dr.iastate.edu/rtd/10077 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. 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Ann Arbor, MI 48106 i Edgeworth expansion and bootstrap approximation for M-estimators of linear regression parameters with increasing dimensions by Muhammad Arif Tiro A Dissertation Submitted to the Graduate Faculty in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY Major: Statistics Approved: Members of the Committee: Signature was redacted for privacy. Signature was redacted for privacy. In Charge of Major Work Signature was redacted for privacy. Signature was redacted for privacy. For the Major Department Signature was redacted for privacy. For the Graduate College Iowa State University Ames, Iowa 1991 Copyright © Muhammad Arif Tiro, 1991. All rights reserved. ii TABLE OF CONTENTS ACKNOWLEDGEMENTS iv 1. INTRODUCTION 1 1.1 Preamble 1 1.2 Statements of the Problems 2 1.3 Basic Notations and Definitions 7 2. LITERATURE REVIEW 11 2.1 Edgeworth Expansion 11 2.2 Bootstrap Approximation ; 19 3. RESULTS ON LEAST SQUARES ESTIMATORS 26 3.1 Introduction 26 3.2 Edgeworth Expansion for Standardized Least Squares Estimators . . 28 3.3 Edgeworth Expansion for Studentized Least Squares Estimators ... 29 3.4 Bootstrap Approximation for Studentized Least Squares Estimators . 33 3.5 Example 34 3.6 Proofs . 36 4. RESULTS ON GENERAL M-ESTIMATORS .59 4.1 Introduction .59 iii 4.2 Edgeworth Expansion for Standardized M-estimators 63 4.3 Edgeworth Expansion for Studentized M-estimators 64 4.4 Bootstrap Approximation for Studentized M-estimators 65 4.5 Example 68 4.6 Proofs 68 5. CONCLUDING REMARKS 82 BIBLIOGRAPHY 85 iv ACKNOWLEDGEMENTS I would like to express my deepest appreciation and sincere gratitude to my major professor Dr. S. N. Lahiri, for his guidance and advice during the course of my research, and to Dr. H. A. David, my major professor during my Master Program and co-major professor for my Ph.D. Program, for various ways he contributed to my education at Iowa State University. It has been a great pleasure having the opportunity to work with and learn from both scholars. Special thanks are due to Dr. V. A. Sposito, Dr. S. Sukhatme, Dr. E. H. Johnston and Dr. R. Hickman, as my committee members, for their valuable contributions to my education. I would like to thank the Indonesian government for providing me with a scholar ship through Indonesian Overseas Training Office - National Building Bureau, during the period of my study at Iowa State University. I am grateful to my mother, for the kind of love, encouragement, and the countless sacrifices she made for me and my family. She has been the foundation of my education, even though my father died before I was born. Finally, I am especially thankful to my wife, Bugiana, my daughters, Rina, Yaya, Ningsih, and Umairah for their patience, tolerance, love and never ending support of me during some difficult times we passed through. Thank you for helping me maintain my perspective and enthusiasm. 1 1. INTRODUCTION 1.1 Preamble The motivation of this study comes from an observation on the simplest kind of asymptotic theory, the one based on normal approximations. In this case we seem to be in paradise, pampered with maximally concentrated estimators, uniformly most powerful tests, and confidence sets which are intervals. There is but one disadvantage: in this paradise, one is rather far from reality. Normal approximation is an oversim plification. Moreover, the normal approximation does not allow one to discriminate between statistical procedures which are equivalent at this level of accuracy. Since the normal approximation reflects the actual performance of statistical procedures only to a certain degree, practical relevant differences may be hidden behind an identical asymptotic behavior. It appears that more refined asymptotic methods - using asymptotic expansions rather than normal approximations - lead to a general statistical theory which, though more complicated than the theory based on normal approximations, is still simple enough to achieve numerical results which are of sufficient accuracy for moderate sample sizes. We hope that asymptotic methods will finally be accepted as solutions to the problems mathematical statistics is faced with. The fact that, in most cases, no other solutions are feasible should support this process.
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