Optimal Design and Related Areas in Optimization and Statistics Book

1 W-Iterations and Ripples Therefrom B. Torsney 1

1.1 Introduction 1

1.2 Optimal Design and an Optimization Problem 1

1.3 Derivatives and Optimality Conditions 2

1.4 Algorithms 3

1.5 A Steepest-Ascent Algorithm 9

1.6 Simultaneous Approach to Optimal Weight and Support Point Determination 9

References 11

2 Studying Convergence of Gradient Algorithms Via Optimal Experimental Design Theory R. Haycroft L. Pronzato H.P. Wynn A. Zhigljavsky 13

2.1 Introduction 13

2.2 Renormalized Version of Gradient Algorithms 14

2.3 A Multiplicative Algorithm for Optimal Design 16

2.4 Constructing Optimality Criteria which Correspond to a Given Gradient Algorithm 18

2.5 Optimum Design Gives the Worst Rate of Convergence 19

2.6 Some Special Cases 20

2.7 The Steepest-Descent Algorithm with Relaxation 23

2.8 Square-Root Algorithm 30

2.9 A-Optimality 32

2.10 [alpha]-Root Algorithm and Comparisons 33

References 36

3 A Dynamical-System Analysis of the Optimum s-Gradient Algorithm L. Pronzato H.P. Wynn A. Zhigljavsky 39

3.1 Introduction 39

3.2 The Optimum s-Gradient Algorithm for the Minimization of a Quadratic Function 40

3.3 Asymptotic Behaviour of the Optimum s-Gradient Algorithm in R[superscript d] 52

3.4 The Optimum 2-Gradient Algorithm in R[superscript d] 55

3.5 Switching Algorithms 66

References 79

4 Bivariate Dependence Orderings for Unordered Categorical Variables A. Giovagnoli J. Marzialetti H.P. Wynn 81

4.1 Introduction 81

4.2 Dependence Orderings for Two Nominal Variables 83

4.3 Inter-Raters Agreement for Categorical Classifications 90

4.4 Conclusions and Further Research 94

References 95

5 Methods in Algebraic Statistics for the Design ofExperiments G. Pistone E. Riccomagno M. P. Rogantin 97

5.1 Introduction 97

5.2 Background 98

5.3 Generalized Confounding and Polynomial Algebra 102

5.4 Models and Monomials 113

5.5 Indicator Function for Complex Coded Designs 118

5.6 Indicator Function vs. Grobner Basis 121

5.7 Mixture Designs 126

5.8 Conclusions 130

References 131

6 The Geometry of Causal Probability Trees that are Algebraically Constrained E. Riccomagno J.Q. Smith 133

6.1 The Algebra of Probability Trees 133

6.2 Manifest Probabilities and Solution Spaces 137

6.3 Expressing Causal Effects Through Algebra 139

6.4 From Models to Causal ACTs to Analysis 142

6.5 Equivalent Causal ACTs 147

6.6 Conclusions 151

References 154

7 Bayes Nets of Time Series: Stochastic Realizations and Projections P.E. Caines R. Deardon H.P. Wynn 155

7.1 Bayes Nets and Projections 155

7.2 Time Series: Stochastic Realization and Conditional Independence 160

7.3 LCO/LCI Time Series 163

7.4 TDAG as Generalized Time 165

References 166

8 Asymptotic Normality of Nonlinear Least Squares under Singular Experimental Designs A. Pazman L. Pronzato 167

8.1 Introduction 167

8.2 The Convergence of the Design Sequence to a Design Measure 171

8.3 Consistency of Estimators 176

8.4 On the Geometry of the Model Under the Design Measure [xi] 179

8.5 The Regular Asymptotic Normality of h([theta superscript N]) 182

8.6 Estimation of a Multidimensional Function H ([theta]) 186

References 190

9 Robust Estimators in Non-linear Regression Models with Long-Range Dependence A. Ivanov N. Leonenko 193

9.1 Introduction 193

9.2 Main Results 196

9.3 Auxiliary Assertions 207

9.4 Proofs 215

References 217

Index 223