Contributions to Statistics

V. Fedorov I W. G. Miiller II. N. Vuchkov (Eds.) Model-Oriented Data Analysis, XII/248 pages. 1992

J. Antoch (Ed.) Computational Aspects of Model Choice, VII I 285 pages, 1993

W. G. Miiller I H. P. Wynn I A. A. Zhigljavsky (Eds.) Model-Oriented Data Analysis XIII/287 pages, 1993

P. Mandl/ M. Hu~kova (Eds.) Asymptotic Statistics X1474 pages, 1994

P. Dirschedll R. Ostermann (Eds.) Computational Statistics VII/553 Pages, 1994

Christos P. Kitsos Werner G. Muller (Eds.)

MODA4-Advancesin Model-Oriented Data Analysis Proceedings of the 4th International Workshop in Spetses, Greece June 5-9, 1995

With 37 Figures

Springer-Verlag Berlin Heidelberg GmbH

Series Editors Wemer A. Miiller Peter Schuster

Editors Dr. Christos P. Kitsos Department of Statistics Athens University of Economics and Business 76 Patission Street GR-Athens 104-34, Greece

Dr. Wemer G. Milller Department of Statistics University of Economics and Business Administration Augasse 2-6 A-1090 Vienna, Austria

ISBN 978-3-7908-0864-3

CIP-Titelaufnahme der Deutschen Bibliothek Advances in model oriented data analysis: proceedings of the 4th international workshop in Spetses, Greece, June 5 - 9, 1995 I MODA 4. Christos P. Kitsos; Werner G. Miiller (ed.).­(Contributions to statistics) ISBN 978-3-7908-0864-3 ISBN 978-3-662-12516-8 (eBook) DOI 10.1007/978-3-662-12516-8

NE: Kitsos, Christos P. [Hrsg.]; MODA <4, 1995, Spetsai>

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© Springer-Verlag Berlin Heidelberg 1995 Originally published by Physica-Verlag Heidelberg in 1995

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PREFACE This volume is the proceedings of the 4th International Workshop on Model-Oriented Data Analysis. This series of events originated in 1987 at a meeting in Eisenach, that successfully brought together scientists from numerous countries of the 'East ' and 'West'. Now that this distinction is obsolete dialogue has been greatly facilitated, providing opportunities for this dialogue, however, is as vital as ever.

The present meeting at Spetses, Greece from 5th to 9th of June 1995 again assembles statisticians from all over the world as this book documents. The hospitality offered by the University of Economics of Athens and the Korgialenios School made it possible to organize this workshop. The editors are also grateful to Intracom (Greece), the Ionian Bank and the Procter & Gamble Company (USA) for their generous support. We would particularly like to mention Dr. Michael Meredith, who being our contact person at Procter & Gamble, enabled us to publish these proceedings. Further thanks go to Dr. Peter Schuster from Physica Verlag Heidelberg for his continuing support of the project.

The contributions to this volume were carefully selected from the submissions by the editors after a one stage refereeing process. We would like to thank the members of the MODA committee, A.C. Atkinson, R.D. Cook, V.V. Fedorov, P.Hackl, H. Lauter, B.Torsney, LN. Vuchkov, H.P.Wynn,and A.A. Zhigljavsky, who not only defined the main topics of the workshop, but also served as the referees.

I. Optimal Design The dominant topic in the submissions was the optimal design of statistical experiments a traditional 'hot' theme at the MODA meetings. The papers of this section deal with such various topics as constrained optimization, Bayesian viewpoint, a.l.gorithms, etc. and cover both branches response surface and categorical designs.

The introductory article by Fedorov- and N achtsheim presents methods for designing experiments when there is a dynamic element in the model. Among other results they succeed in illuminating the close relations to marginally constrained designs. Montepiedra also makes use of techniques originally developed for constrained designs by applying them for the purpose of model robustification. As Clyde shows in a Bayesian context constrained designs can also be applied to improve normal approximation in statistical models. The Bayesian perspective is maintained by P. Muller and Parmigiani when they provide illuminative 'real life' examples of one stage and sequential optimal designs. A different sequential design strategy builds the framework in which Galtchouk and Maljutov investigate the lower bound of its duration.

The next couple of papers cover optimal designs for various criteria and types of re­sponses. For a generalized minimax criterion in the simple linear regression model Torsney and Fidalgo characterize optimal designs with not more than two points of support. The D-criterion is referred to by Kitsos who calculates support points of optimal designs for various nonlinear models used in chemical kinetics. That product designs are D-optimal in additive nonlinear models is demonstrated by Schwabe. For generalized linear models the properties of D-optimal designs are discussed along with a parameter dimensionality reduction technique by Sitter and Torsney.

Hilgers examines properties of generalized tic polynomials on the simplex that have applications in mixture experiments, e.g. in chemistry. Weighing designs which are treated by Ceranka and Katulska have applications in the same field. The question of whether the

VI

risk of loosing observations results in dramatic changes of the optimal design is investigated by Hackl. In contrast, the model robustness of designs is the aim of the contribution by Ibrahimy and Cook. They consider models where the form of the conditional distribution of the response on the linear predictor is unknown.

The next block of papers is related to categorical models, Koukouvinos presents a col­lection of new D-optimal first order saturated designs with n = 2mod4 observations. Block designs are considered in the three papers that follow by Bogacka, Ceranka and Kacmarek, and Mejza and Kageyama. The first gives a discussion of information matrices in gener­ally balanced block designs, the second treats a problem from crossing experiments, the parameter estimation in factorial triallel analysis, and the third gives statistical properties of certain nested block designs. This latter provides an extension of the ideas of optimality to random effects models.

The concluding papers of the section by Donev and Jones, and W.G. Miiller and Zimmerman devote themselves to algorithms for finding optimal designs. The former describes a flexible algorithm for the construction of optimum cross-over designs, whereas the latter attempts to overcome the problem of correlated observations by a suitable separation technique.

II. Estimation and Optimization Atkinson provides an extensive discussion of diagnostic techniques for the effect of indi­vidual observations on multivariate transformations to normality and adds a collection of illuminating examples. Jureckova extends her results on the uniform asymptotic linearity of regression rank scores to a broader class of error distributions. Further results on the asymptotic distribution of regression parameters and a corresponding simulation experi­ment are offered by Parring. Droge also gives simulation results on the behaviour of model selection criteria under cross validation.

The following two papers are concerned with robust estimation. The first of these papers by Ch. Muller and Kitsos discusses the asymptotics of one-step M-estimators and gives applications from rhythmometry. The second by Shevlyakov and Vil'chevskiy provides asymptotic and finite sample properties of adaptive minimax M-estimators. A Bayesian perspective to heterogeneity and overdispersion via Gibbs sampling is offered by Dey, Peng and Larose. This perspective is illustrated by an example on toxoplasmosis. Gibbs sampling is also employed by Polasek and P. Miiller for the estimation of volatility models in finance.

A recursive algorithm based on stochastic approximation and kernel estimation accom­panied by an application from chemical engineering is introduced by G.Yin and K.Yin. A similar application field might apply to the work of Vuchkov and Boyadijeva, who use opti­mization techniques for quality improvement of processes which depend upon quantitative and qualitative factors and describe the effect of different target locations. Optimization is also central to the final paper by Pronzato, Wynn and Zhigljavsky who improve the Golden-section algorithm for locally symmetric functions by renormalization.

It is our hope that the present collection proves to be of interest not only for the participants of the meeting but also for a greater range of scientists from the particular subject fields.

C.P. Kitsos and W.G. Miiller Vienna, January 1995

Contents

I Optimal Design

Optimal Designs for Time-Dependent Responses

VALERY FEDOROV and CHRIS NACHTSHEIM

1 Introduction . . . . . . . . .

2 Model and Design Criterion

3 Analytic Approaches . . . .

4 Trajectory Parameterization .

5 Generalized Trajectories as Conditional Distributions .

6 Algorithms . . . . . . . . . . . . . . . . . . .

7 Extension: Linear Functions of the Response

Robust Optimal Designs with Constraints

GRACE MONTEPIEDRA

1 Introduction ....

2 Convex Criteria with Convex Constraints

3 The D- and A-restricted Minimum Bias Designs.

4 Examples .....

5 Appendix: Proofs .

Bayesian Designs for Approximate Normality

MERLISE A. CLYDE

1 Introduction . .

2 Curvature Measures

3 Designs for Approximate Normality

4 Rumford Example . . . . . . . . . .

4.1 Minimum Curvature Designs

4.2 Constrained Designs

5 Discussion . . . . . . . . . .

1

3

3

3

5

7

8

10

12

15

15

17

18

20

22

25

25

26

27

28

28

29

33

Vlll

Simulation Approach to One-Stage and Sequential Optimal Design Prob-lems 37

GIOVANNI PARMIGIANI and PETER MULLER

1

2

3

4

Introduction ........ .

One-Stage Design Problems

Examples for One-Stage Optimal Designs

3.1

3.2

3.3

Example 1: An Information Theoretic Stopping Rule .

Example 2: Timing of Medical Exams . . . . . . . .

Example 3: Optimal Design for Heart Defibrillators

Sequential Problems . . . . . . . . . . . . . . . . . . . . . .

37

38

39

39

40

41

42

4.1 Sequential Optimal Design by Smoothing of Monte Carlo Experiments 42

4.2 Example 4: Optimal Sequential Design for a Logistic Growth Model 44

One Bound for the Mean Duration of Sequential Testing Homogeneity 49

LEONID I. GALTCHOUK and MICHAIL B. MALJUTOV

1

2

Introduction and Main Results . . . . . . . . . .

Maximum of the Lower Bound (2) for the CaseD 1.

MY-optimization in Simple Linear Regression

BEN ToRSNEY and JEsus LOPEZ-FIDALGO

1 Introduction . . . . . . . . .

2 Solution for Different Cases

2.1 Case1

2.2 Case2

2.3 Case3

3 Conclusion

4 A Note on Correlation

On the Support Points of D-Optimal Nonlinear Experimental Designs

49

53

57

57

60

60

60

61

64

68

for Chemical Kinetics 71

CHRISTOS P. KITSOS

1

2

3

Introduction .

Background .

Support Points for D-Optimal Designs

71

72

73

Designing Experiments for Additive Nonlinear Models

RAINER SCHWABE

1 Introduction . ........

2 Additive Nonlinear Models

3 A Complete Class Result

4 Locally Optimal Designs .

5 Minimax Optimal Designs

6 Weighted Optimality

7 Applications . . . . .

D-Optimal Designs for Generalized Linear Models

RANDY R. SITTER and BEN TORSNEY

1

2

3

4

5

6

Introduction . . . . . . . . . . . . .

A Canonical Form for Generalized Linear Models

The Geometry of G .

D-Optimal Designs .

Some Examples

Discussion . . .

ix

77

77

79

79

80

81

82

83

87

87

88

89

93

95

. 101

Further Results on Optimal Designs for Generalized Tic Polynomials on the Simplex 103

RALF-DIETER HILGERS

1

2

3

4

5

Introduction .

Preliminaries

!pp-optimality for Generalized Tic Polynomials

Examples for Generalized Tic Polynomials without Linear Terms

Further Comments . . . . . . . . . . . . . . . . . . . . . . . . . .

. 103

. 104

. 105

. 107

. 108

Relations between Spring and Chemical Balance Weighing Designs with the Diagonal Covariance Matrix of Errors 111

BRONISLAW CERANKA and KRYSTYNA KATULSKA

1

2

3

Introduction . . . . . . . . .

Optimum Weighing Designs

Some Optimum Weighing Designs

. 111

. 112

. 114

X

Optimal Design for Experiments with Potentially Failing Trials 117

PETER HACKL

1

2

3

4

Introduction .

Statement of the Problem

Examples ........ .

3.1

3.2

3.3

A Simple Probability Structure

Independently Failing Trials . .

Increasing Probability for Failing Trials

Concluding Remarks .............. .

Regression Design for One-Dimensional Subspaces

ABDELOUAFI IBRAHIMY and R. DENNIS CooK

1

2

3

4

5

6

7

Introduction . .

Design Criteria

Equally Spaced Designs

A Two-Dimensional Example

Constructing Equally Distributed Designs

5.1

5.2

5.3

In 2 and 3 Dimensions

In p Dimensions . . .

ESD's and Rotatability

Discussion . . . . . . . . . . . .

Appendix: Characterization of Equally Distributed Designs

. 117

. 118

. 120

. 120

. 121

. 122

. 123

125

. 125

. 126

. 127

. 129

. 130

. 130

. 131

. 131

. 132

. 132

D-Optimal First Order Saturated Designs with n = 2mod4 Observations135

CHRISTOS KOUKOUVINOS

1

2

3

Introduction . . . . .

D-Optimal Designs for n = 2mod4

Some New D-Optimal Designs ...

. 135

. 136

. 137

On Information Matrices for Fixed and Random Parameters in Gener-

ally Balanced Experimental Block Designs 141

BARBARA BoGACKA

1

2

3

The Model ...

General Balance of the Designs

Information Matrices of the Design

. 141

. 142

. 144

xi

Estimation of Parameters in Factorial Triallel Analysis for BIB Design - the Mixed Model 151

BRONISLAW CERANKA and ZYGMUNT KACZMAREK

1

2

3

Introduction . . . . . .

Model of Observations

Estimation of Parameters

. 151

. 152

. 154

On the Optimality of Certain Nested Block Designs under a Mixed Ef-fects Model 157

STANISLAW MEJZA and SANPEI KAGEYAMA

1

2

3

4

Introduction . . . .

The Linear Model

Optimality Criteria .

Characterization of D( v, Rb, k)

Construction of A-Optimum Cross-Over Designs

ALEXANDER N. DONEY and BYRON JONES

1

2

3

4

Introduction . . . . . . . . . . . . . . . .

Modelling Assumptions and Criterion of Optimality

An Algorithm . . . . . . .

Examples and Discussion

. 157

. 158

. 159

. 161

165

. 165

. 166

. 168

. 168

An Algorithm for Sampling Optimization for Semivariogram Estimation173

WERNER G. MULLER and DALE L. ZIMMERMAN

1 Introduction . . . . . . . . . .... . . . 173

2 The Semivariogram and Its Estimation . . 174

3 Design Considerations . 175

4 An Algorithm . 176

5 The Example . 177

6 Conclusions . 178

II Estimation and Optimization 179

Multivariate Transformations, Regression Diagnostics and Seemingly Unrelated Regression 181

ANTHONY C. ATKINSON

1

2

Introduction . . . . .

Multivariate Transformations to Normality

. 181

. 182

xu

3

4

5

6

7

8

Deletion Statistics for the Likelihood Ratio Test

Constructed Variables and Seemingly Unrelated Regression

Examples of Seemingly Unrelated Regression Diagnostics

Structure of the Seemingly Unrelated Regression Test

Regression .

Discussion .

. 183

. 184

. 187

. 188

. 190

. 190

Regression Rank Scores: Asymptotic Linearity and RR-Estimators 193

JANA JURECKOVA

1

2

3

Introduction .

RR-Estimators

Uniform Asymptotic Linearity of Regression Rank Scores Process .

The Asymptotic Distribution of Regression Parameters

ANNE-MAl PARRING

1 The Problem . . .........

2 The Linear Regression Function .

3 The Distribution of LS Estimates

4 The Example ........

5 The Simulation Experiment

6 Conclusion ..........

. 193

. 195

. 197

205

. 205

. 206

. 207

. 209

. 210

. 212

Some Simulation Results on Cross-Validation and Competitors for Model Choice 213

BERND DROGE

1

2

3

Introduction .

Model Selection Criteria

Simulations . . . . . . .

Robust Estimation of Non-linear Aspects

CHRISTOS P. KITSOS and CHRISTINE H. MULLER

. 213

. 214

. 216

223

1 Introduction .................................... 223

2 Robust Estimators for Non-linear Aspects and their Asymptotic Behaviour 225

3

4

5

Efficiency of Robust Estimators and Designs .

Example: The Rhythmometry Problem

Discussion . . . . . . . . . . . . . . . . .

. 226

. 228

. 231

xiii

Robust Minimax Adaptive M-Estimators of Regression Parameters 235

GEORGIY L. SHEVLYAKOV and NIKITA 0. VIL'CHEVSKIY

1

2

3

4

Introduction . . . . . . . . . . . .

Robust Minimax M-Estimators .

The Least Favorable Density for the Class :F12

Robust Adaptive M-estimators ........ .

. 235

. 236

. 237

. 238

Modeling Heterogeneity and Extraneous Variation Using Weighted Dis-tributions 241

DIPAK K. DEY , FENGCHUN PENG and DANIEL LAROSE

1

2

3

4

Introduction . . . . . . . . . . . . . . . . .

Overdispersed Models for Clumped Data .

Prior Specification for Overdispersed Models

An Overdispersed Binomial Model . . . . . .

Gibbs Sampling for ARCH Models in Finance

WoLFGANG PoLASEK and PETER MULLER

1

2

Introduction ........... .

1.1 ARCH Models in Finance

The Univariate ARCH Model

2.1

2.2

Estimation . . . . . .

The Full Conditional and Probing Distributions .

. 241

. 243

. 245

. 246

251

. 251

. 252

. 252

. 253

. 253

2.3 The Independence Chain - Hastings Algorithm . . 254

3 Example: The Lag1 US/DM Volatility Model (Univariate Estimation Results)254

4

5

The Multivariate ARCH Model ............. .

4.1 The Vector ARCH Matrix Probing Distribution .

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . .

. 256

. 257

. 259

A Class of Recursive Algorithms Using Non-parametric Methods with Constant Step Size and Window Width: A Numerical Study 261

GEORGE YIN and KEWEN YIN

1 Introduction . . . ... . 261

2 Recursive Algorithms . . 262

3 Main Results .. . . . . 263

4 Simulation and Numerical Experiments . 264

5 Further Discussions . . . ~ .. . . . .... . 268

xiv

Robust Design of Products Depending on Both Qualitative and Quan-titative Factors 273

IVAN N. VucHKOV and L. N. BoYADJIEVA

1 Introduction .................................... 273

2 Models of Performance Characteristic Depending on Both Qualitative and Quantitative Factors ............................... 274

3 Models of Performance Characteristic's Mean Value a.nd Variance in Mass

4

Production . . . . . . . . . 276

Optimization Procedures . . .

4.1 Problem Formulation

. 277

. 277

4.2 Optimization when the Target Coincides with the Extremum of the Performance Characteristic . . . . . . . . . . . . . . . . . . . . . . . 278

4.3 Optimization when the Target does not Coincide with the Extremum of the Performance Characteristic . 279

5 Example ...................................... 280

Improving on Golden-Section Optimisation for Locally Symmetric Func-tions 285

Luc PRONZATO, HENRY P. WYNN and ANATOLY A. ZHIGLJAVSKY

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 Asymptotic Behaviour of the Golden Section Algorithm for Loca.lly Sym­metric Functions . . . . . . . . . . . . . .

3 The Dynamic-Programming Point of View

. 285

. 288

. 292