Edgeworth expansion and bootstrap approximation for M-estimators of linear regression

97 Pages·2017·1.39 MB·English

by  Muhammad Arif Tiro

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
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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
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Order Number 9202398
Edgeworth expansion and bootstrap approximation for
M-estimators of linear regression parameters with increasing
dimensions
Tiro, Muhammad Arif, Ph.D.
Iowa State University, 1991
U M I
300N.ZeebRd.
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.