Topics in Optimal Design

This book is a follow up of 'Theory of Optimal Designs' by K.R. Shah and Bikas K. Sinha published in this series. Unlike the previous book, this one covers a wide range of topics in both discrete and continuous optimal designs. The topics discussed include designs for regression models, covariates models, models with trend effects, and models with competition effects. An extensive study on a new optimality criterion is also presented. The prerequisites are a basic course in the design and analysis of experiments and some familiarity with the concepts of optimality criteria. Erkki P. Liski is Professor of Statistics at the University of Tampere in Finland. He has been head of various research projects on theory and applications of statistics with the Academy of Finland. Nripes K. Mandal is Professor of Statistics at Calcutta University in India. He is closely associated with all academic activities of the Calcutta Statistical Association. Kirti R. Shah is Professor of Statistics at the University of Waterloo in Canada. He is currently the president of the International Indian Statistical Association. Bikas K. Sinha is Professor of Statistics at the Indian Statistical Institute in Calcutta, India. He has served on the editorial board of a number of international statistical journals.

Erkki P. Liski is Professor of Statistics at the University of Tampere, in Finland.

Preface

v

Scope of the Monograph

1

(20)

Introduction and Literature Review

1

(2)

Some Useful Linear Models

3

(6)

A Polynomial Regression Model: Exact and Approximate Theory

4

(2)

A General Mixed Linear Model

6

(1)

A Random Coefficient Regression (RCR) Model

7

(1)

A Growth Curve Formulation

8

(1)

Estimation of Regression Parameters in RCR Models

9

(2)

Chapter Summary

11

(10)

References

14

(7)

Optimal Regression Designs in Symmetric Domains

21

(16)

Introduction

22

(1)

Loewner Comparison of Designs

22

(2)

Polynomial Fit Models

24

(3)

Symmetric Polynomial Designs

25

(1)

Symmetric Designs for Quadratic Regression

25

(2)

Multi-Factor First-Degree Polynomial Fit Models

27

(10)

Designs in a Euclidean Ball

28

(4)

Designs in a Unit Hypercube

32

(4)

References

36

(1)

Optimal Regression Designs in Asymmetric Domains

37

(34)

Introduction

38

(3)

de la Garza Phenomenon in Quadratic and Cubic Regression

41

(11)

Quadratic Regression with Full Set of Parameters

42

(4)

Quadratic Regression without the Intercept Term

46

(4)

Optimal Designs in a Cubic Regression Model

50

(2)

Optimal Designs for Parameter Estimation in RCR Models

52

(6)

First-Degree RCR Model

52

(2)

Quadratic RCR Model

54

(4)

Optimal Designs for Prediction in RCR Models

58

(4)

Introduction

58

(1)

First-Degree RCR Model

59

(2)

Quadratic RCR Model

61

(1)

Optimal Designs for Inverse Prediction in RCR Models

62

(9)

First-Degree Regression Model

63

(2)

Quadratic Regression

65

(4)

References

69

(2)

Optimal Designs for Covariates' Models with Structured Intercept Parameter

71

(24)

Introduction

72

(1)

Optimal Regression Designs with One-Way Classified Intercepts

73

(8)

Optimal Regression Designs with Two-Way Classified Intercepts

81

(10)

Optimal Regression Designs in an RBD Set-Up

82

(1)

Constructional Procedures when b and v Are Even Integers and c* = 1

83

(1)

Constructional Procedures for b = v = 0 (mod 4)

83

(3)

Miscellaneous Results

86

(1)

Optimum Choice of Covariates in a BIBD Set-Up

87

(4)

Concluding Remarks

91

(4)

References

92

(3)

Stochastic Distance Optimality

95

(20)

Introduction

95

(3)

Properties of the DS-Optimality Criterion

98

(3)

Isotonicity and Admissibility

99

(1)

Schur-Concavity of the DS-Optimality Criterion

99

(2)

Discrete DS-Optimal Designs

101

(5)

Treatment vs. Control Designs

102

(2)

Weighted Coverage Probabilities

104

(2)

DS-Optimal Regression Designs

106

(5)

First-Degree Polynomial Fit Models

107

(2)

Symmetric Polynomial Designs

109

(1)

Characterization of D- and E-Optimality Using DS(ε)-Criterion

110

(1)

Generalizations of DS-Optimality

111

(4)

References

113

(2)

Designs in the Presence of Trends

115

(12)

Introduction

115

(1)

Preliminaries

116

(1)

Optimality within Restricted Classes

117

(3)

Optimal Designs in D(v, b, k)

120

(5)

Efficiency Bounds

125

(2)

References

125

(2)

Additional Selected Topics

127

(30)

Introduction

127

(2)

Optimal Designs for a Competing Effects Model

129

(3)

Split Block Designs

132

(5)

Nested Experimental Designs

137

(5)

Introduction

137

(1)

Nested Block Designs

138

(3)

Optimal Nested Block Designs

141

(1)

Nested Row-Column Designs

142

(1)

Optimality Status of Incomplete Layout Three-Way Balanced Designs

142

(6)

Three-Way Balanced Designs Based on Agrawal's Method 1

144

(1)

Three-Way Balanced Designs Based on Agrawal's Method 3

145

(1)

Three-Way Balanced Designs Based on Agrawal's Method 4

146

(1)

Concluding Remarks

147

(1)

Optimal Designs Under Heteroscedastic Errors in Linear Regression

148

(9)

A Model with Heteroscedastic Errors

148

(1)

Complete Class of Designs in Minkin's Set-Up

149

(2)

Specific Optimal Designs

151

(2)

References

153

(4)

Author Index

157

(4)

Subject Index

161

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