Table Of ContentLecture Notes in Statistics 124
Edited by P. Bickel, P. Diggle, S. Fienberg, K. Krickeberg,
I. OIkin, N. Wermuth, S. Zeger
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Christine H. Muller
Robust Planning and Analysis of
Experiments
, Springer
Christine H. MOller
Freie Universitat Berlin
Fachbereich Mathematik und Informatik, WEI
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To my mother
Preface
Up to now the two different areas of" Optimal Design of Experiments" and
"Robust Statistics", in particular of "Outlier Robust Statistics", are very
separate areas of Statistics. There exist several books on optimum experi
mental design like those of Fedorov (1972), Silvey (1980), Pazman (1986),
Shah and Sinha (1989), Atkinson and Donev (1992), Pukelsheim (1993) and
Schwabe (1996a). There exist also several books on robust statistics like
those of Huber (1981), Hampel et al. (1986), Tiku et al. (1986), Rousseeuw
and Leroy (1987), Kariya and Sinha (1988), Staudte and Sheather (1990),
Biining (1991), Rieder (1994) and Jureckova and Sen (1996). But there is
almost no overlapping between the books on optimum experimental design
and on robust statistics. Now the presented book will give a first link be
tween these two areas. It will show that a robust inference will profit from
an optimal design and that an optimal design is more reasonable if it allows
also a robust analysis of the data.
The first part of the presented book gives an overview on the foundations
of optimum experimental design. In this classical approach a design is op
timal if the efficiency of the least squares estimator (or the classical F -test)
is maximized within all possible designs. But the least squares estimators
and the F-tests are not robust against outliers. In the presence of outliers
they can be biased very much.
Outlier robust estimators and tests are derived in the second part of
the book which provides the foundations of outlier robust statistics espe
cially for planned experiments. This already differs very much from the
approaches in the books and the majority of the papers on robust statis
tics because they usually do not regard planned experiments. The majority
of the approaches in robust statistics 3.'lsumes that the independent vari
ables in regression experiments are random so that outliers can appear also
in these independent variables. For planned experiments and for models
with qualitative factors these assumptions make no sense so that the ex
isting main concepts of outlier robust inference have to be specified for
planned experiments. This is done in the second part of the book. As main
qualitative concepts of outlier robustness the continuity and Frechet dif
ferentiability of statistical functionals are regarded. From these qualitative
concepts the quantitative concepts of outlier robustness are derived. These
quantitative concepts are the breakdown point and the bias in shrinking
contamination neighbourhoods, which is closely related to Hampel's con
cept based on influence functions. These concepts are mainly explained
for general linear models but also their meaning for nonlinear problems is
discussed.
Preface
Vlll
In the third part the efficiency and robustness of estimators and tests
are linked and the influence of the designs on robustness and efficiency is
studied. For the designs this leads to new nontrivial optimality problems.
These new optimality problems are solved for several situations in the third
part which consists of Section 7, 8 and 9.
In Section 7 and 8 it is shown that the classical A- and D-optimal de
signs provide highest robustness and highest efficiency under robustness
constraints if the robustness measure is based on the bias in shrinking
contamination neighbourhoods. Thereby it turns out that the most robust
estimators and tests and the most efficient robust estimators and tests have
a very simple form at the classical optimal designs. Moreover, most robust
tests and most efficient robust test can be only characterized at the D
optimal designs. Similar results also hold if a nonlinear aspect should be
estimated or if the model is nonlinear. All these results show that the ro
bust statistical analysis profits very much from an optimal choice of the
design.
While in the robustness concept based on shrinking contamination the
classical optimal designs are also optimal for the robust inference, the op
posite is true for the robustness measure based on the breakdown point.
This is demonstrated in Section 9. There ,it is shown that the designs which
are optimal with respect to a high breakdown point are in general very dif
ferent from the classical optimal designs. Because of this difference it is also
discussed how to find a design which combines high breakdown point and
high efficiency of the estimators.
The problem of combining high breakdown point and high efficiency
provides for the designs plenty of new optimality problems which are not
solved up to now. Besides these open problems there are also many other
open problems for further research which are discussed in the outlook.
Finally, I would like to express my thanks to all who supported me in fin
ishing the present work. Above all I am indebted to Prof. Dr. V. Kurotschka
for his fruitful discussions and criticism. Thanks also to all my colleagues,
in particular to Dr. W. Wierich, who teached me a lot of statistics, and to
Dr. R. Schwabe, who was always very co-operative. I also thank Prof. Dr.
H. Rieder for providing some preprints which initiated the present work.
Moreover, I am particularly thankful to the corresponding editor J. Kim
mel of the Lecture Notes ill Statistics and to the referees for very valuable
comments.
At last I would like to thank my family and in particular my husband and
my mother for their support. Especially, I am very grateful to my mother
because without her engaged care for my children this work would not have
been possible.
Christine H. Miiller, Berlill, March 1997
Contents
Preface Vll
Part I: Efficient Inference for Planned Experiments 1
1 Planned Experiments 2
1.1 Deterministic and Random Designs. 2
1.2 Linear and Nonlinear Models 3
1.3 Identifiability of Aspects . . . . . . . 4
2 Efficiency Concepts for Outlier-Free Observations 10
2.1 Assumptions on the Error Distribution. . 10
2.2 Optimal Inference for Linear Problems. . 11
2.3 Efficient Inference for Nonlinear Problems 19
Part II: Robust Inference for Planned Experiments 25
3 Smoothness Concepts of Outlier Robustness 26
3.1 Distributions Modelling Outliers . . .... 26
3.2 Smoothness of Estimat.ors and Functionals 32
3.3 Frechet Differentiabilit.y of M-Fullctionals . 39
4 Robustness Measures: Bias and Breakdown Points 48
4.1 Asympt.otic Bias and Breakdown Points .... 48
4.2 Bias and Breakdown Points for Finite Samples 52
4.3 Breakdown Points in Linear Models ... . 57
4.4 Breakdown Points for Nonlinear Problems .. . 68
5 Asymptotic Robustness for 'Shrinking Contamination 75
5.1 Asymptotic Behaviour of Estimators in Shrinking
Neighbourhoods . . ................. . 75
5.2 Robust Estimation in Contaminated Linear Models. 84
5.3 Robust Estimation of Nonlinear Aspects ...... . 92
5.4 Robust Estimation in Contaminated Nonlinear Models. 94
6 Robustness of Tests 98
6.1 Bias and Breakdown Points 98
6.2 Asymptotic Robustness for Shrinking Contamination. 100
x Contents
Part III: High Robustness and High Efficiency 109
7 High Robustness and High Efficiency of Estimation 110
7.1 Estimators and Designs with Minimum Asymptotic Bias. 110
7.2 Optimal Estimators and Designs for a Bias Bound . . 118
7.3 Robust and Efficient Estimation of Nonlinear Aspects 133
7.4 Robust and Efficient Estimation in Nonlinear Models 139
8 High Robustness and High Efficiency of Tests 143
8.1 Tests and Designs with Minimum Asymptotic Bias 143
8.2 Optimal Tests and Designs for a Bias Bound 153
9 High Breakdown Point and High Efficiency 171
9.1 Breakdown Point Maximizing Estimators and Designs 171
9.2 Combining High Breakdown Point and High Efficiency . 178
Outlook 184
Appendix 189
A.l Asymptotic Linearity of Frechet Differentiable Functionals 189
A.2 Properties of Special Matrices and Functions . . . . . . .. 193
References 207
List of Symbols 225
Index 231
