Contents
Preface vii
1. Preliminaries on probability 1
2. Generalities about the conventional theory of design of experiments . 3
3. Optimal sample size 5
4. Preliminaries on regression 8
5. Design for linear regression: Elfving's method 13
6. Maximum-likelihood estimation 16
7. Locally optimal designs for estimation 23
8. More design in regression experiments 31
8.1. Game theoretic considerations 32
8.2. Some properties of determinants 35
8.3. Proof of Theorem 8.1 36
9. Testing hypotheses 37
9.1. Large sample considerations 42
9.2. Kullback-Leibler information numbers 48
9.3. A multivariate normal example 50
9.4. Consistency of the maximum-likelihood estimate 52
10. Optimal sample size in testing 55
11. Sequential probability-ratio test 57
12. Optimality of sequential probability-ratio test 63
13. Motivation for an approach to sequential design of experiments in
testing hypotheses 66
14. Asymptotic optimality of Procedure A in sequential design 70
15. Extensions and open questions in sequential design 72
16. The problem of adjacent hypotheses 75
17. Testing for the sign of a normal mean: No indifference zone 84
17.1. Posterior distributions and risks 85
17.2. Stopping problems 87
17.3. Continuous time stopping problems 88
17.4. Continuous time problems: Relevance of stopping sets 90
17.5. Stopping problems: Relevance of the heat equation and the free
boundary problem 92
17.6. Bounds and asymptotic expansions for solutions 95
17.7. Discrete time problems revisited 100
18. Bandit problems 102
19. Sequential estimation of a normal mean 108
20. Sequential estimation of the common mean of two normal populations 111
21. Conclusion 117
References 117