Functional Approach to Optimal Experimental Design

by Melas, Viatcheslav B.; Melas, V. B.

• ISBN13: 9780387987415
• ISBN10: 038798741X
• Format: Paperback
• Copyright: 2006-01-01
• Publisher: Springer Verlag

Summary

The subject of the book is a functional theory of optimal designs elaborated by the author during the last two decades. This theory relates to points and weight of optimal designs considered as functions of some values. For linear models these values are metric characteristics of the set of admissible experimental conditions, for example, the bounds of a segment. For nonlinear models they are true values of the parameter to be estimated. Particularly locally D- optimal designs for exponential regression as an important example of nonlinear models and E-optimal designs for polynomial regression on arbitrary segments will be fully studied.

Author Biography

Viatcheslav B. Melas is Professor of Statistics and Numerical Analysis at the St. Petersburg State University.

Table of Contents

Preface	v
Introduction	1	(4)
Fundamentals of the Optimal Experimental Design	5	(18)
The Regression Equation	5	(1)
Gauss-Markov Theorem	6	(1)
Experimental Designs and Information Matrices	7	(2)
Optimality Criteria	9	(2)
D-Criterion	9	(1)
G-Criterion	10	(1)
MV-Criterion	10	(1)
c-Criterion	10	(1)
E-Criterion	11	(1)
Equivalence Theorems	11	(1)
Iterative Numerical Techniques	12	(1)
Nonlinear Regression Models	13	(4)
The Implicit Function Theorem	17	(1)
Chebyshev Models	18	(5)
The Functional Approach	23	(48)
Introduction	23	(2)
Basic Ideas of the Functional Approach	25	(19)
Exponential regression models	25	(4)
Locally D-optimal designs	29	(10)
Maximin efficient designs	39	(5)
Description of the Model	44	(5)
Assumptions and notation	45	(1)
The basic equation	46	(2)
The uniqueness and the analytical properties	48	(1)
The Study of the Basic Equation	49	(11)
Properties of implicit functions	49	(2)
Jacobian of the basic equation	51	(2)
On the representation of implicit functions	53	(2)
The monotony property	55	(5)
Three-Parameter Logistic Distribution	60	(3)
Appendix: Proofs	63	(8)
Proof of Theorems 2.4.2, 2.4.3, and 2.4.4	63	(2)
Proof of Theorem 2.3.1	65	(2)
Proof of Theorem 2.2.3	67	(4)
Polynomial Models	71	(64)
Introduction	71	(2)
Designs for Individual Coefficients	73	(14)
Statement of the problem	73	(1)
ek-Optimal designs	74	(1)
Analytical properties of ek-optimal designs	75	(7)
A numerical example	82	(5)
E-Optimal Designs: Preliminary Results	87	(17)
Statement of the problem and a dual theorem	88	(1)
The number of support points	89	(4)
Chebyshev designs	93	(10)
A boundary equation	103	(1)
Non-Chebyshev E-Optimal Designs	104	(31)
Basic equation	104	(10)
Limiting designs	114	(7)
Proof of the main theorem	121	(3)
Examples	124	(11)
Trigonometrical Models	135	(62)
Introduction	135	(2)
D-Optimal Designs	137	(18)
Preliminary results for D-optimal designs	137	(5)
Analytic properties of D-optimal designs	142	(6)
The differential equation and the eigenvalue problem	148	(2)
A functional-algebraic approach	150	(3)
Examples	153	(2)
E-Optimal Designs	155	(37)
Preliminary results and E-optimal designs on large design spaces	155	(5)
E-optimal designs on sufficiently small intervals	160	(12)
Example: The linear trigonometric regression model on a partial circle	172	(2)
E-Optimal designs on arbitrary intervals	174	(18)
Numerical Comparison of D- and E-Optimal Designs	192	(5)
D-Optimal Designs for Rational Models	197	(20)
Introduction	197	(1)
Description of the Model	198	(2)
The Number of Points	200	(3)
Optimal Design Function	203	(2)
Algebraic Approach and Limiting Designs	205	(8)
The Taylor Expansion	213	(4)
D-Optimal Designs for Exponential Models	217	(14)
The Number of Support Points	218	(6)
Optimal Design Function	224	(4)
Taylor Expansions	228	(3)
E- and c-Optimal Designs	231	(42)
Introduction	231	(3)
Preliminary Results	234	(4)
Asymptotic Analysis of E- and c-Optimal Designs	238	(5)
Analytical Properties of Optimal Designs	243	(2)
Rational Models	245	(14)
Exponential Models	259	(9)
Appendix: Some Auxiliary Results	268	(5)
The Monod Model	273	(44)
Introduction	273	(3)
Equivalent Regression Models	276	(6)
Locally D-Optimal Designs	282	(3)
A Numerical Study	285	(4)
Taylor Expansions	289	(5)
Locally E- and ek-Optimal Designs	294	(4)
Maximin Efficient Designs	298	(8)
A numerical procedure	299	(2)
A comparison of maximin and uniform designs	301	(5)
Appendix	306	(11)
Proof of Lemma 8.2.1	310	(1)
Proof of Theorem 8.2.1	310	(2)
Proof of Lemma 8.3.1	312	(1)
Proof of Lemma 8.3.2	313	(2)
Proof of Lemma 8.6.1	315	(2)
Appendix	317	(4)
References	321	(10)
Index	331