Mathematical Theory of Statistics: Statistical Experiments and Asymptotic Decision Theory

✍ Scribed by Helmut Strasser

• Publisher: De Gruyter
• Year: 1985
• Tongue: English
• Leaves: 504
• Series: De Gruyter Studies in Mathematics; 7
• Edition: Reprint 2011
• Category: Library

✦ Synopsis

"The book is well written but it is not for a beginner. It will be quite useful for workers doing research in the area of asymptotic theory of statistical inference." Mathematical Reviews

"A very interesting feature of the book is the introductory remark at the beginning of each chapter which indicates the historical as well as conceptual development of the material presented in that chapter. [...] This is a very well written, tough (at least to the reviewer) mathematical text and would be extremely useful to the research workers interested in asymptotics in general. [...] The author must be congratulated on a job well done and it is hoped that this text may attract many more able statisticians to probe deeper into these problems." Zentralblatt für Mathematik

✦ Table of Contents

Chapter 1: Basic Notions on Probability Measures
1. Decomposition of probability measures
2. Distances between probability measures
3. Topologies and σ-fields on sets of probability measures
4. Separable sets of probability measures
5. Transforms of bounded Borel measures
6. Miscellaneous results
Chapter 2: Elementary Theory of Testing Hypotheses
7. Basic definitions
8. Neyman-Pearson theory for binary experiments
9. Experiments with monotone likelihood ratios
10. The generalized lemma of Neyman-Pearson
11. Exponential experiments of rank 1
12. Two-sided testing for exponential experiments: Part 1
13. Two-sided testing for exponential experiments: Part 2
Chapter 3: Binary Experiments
14. The error function
15. Comparison of binary experiments
16. Representation of experiment types
17. Concave functions and Mellin transforms
18. Contiguity of probability measures
Chapter 4: Sufficiency, Exhaustivity, and Randomizations
19. The idea of sufficiency
20. Pairwise sufficiency and the factorization theorem
21. Sufficiency and topology
22. Comparison of dominated experiments by testing problems
23. Exhaustivity
24. Randomization of experiments
25. Statistical isomorphism
Chapter 5: Exponential Experiments
26. Basic facts
27. Conditional tests
28. Gaussian shifts with nuisance parameters
Chapter 6: More Theory of Testing
29. Complete classes of tests
30. Testing for Gaussian shifts
31. Reduction of testing problems by invariance
32. The theorem of Hunt and Stein
Chapter 7: Theory of estimation
33. Basic notions of estimation
34. Median unbiased estimation for Gaussian shifts
35. Mean unbiased estimation
36. Estimation by desintegration
37. Generalized Bayes estimates
38. Full shift experiments and the convolution theorem
39. The structure model
40. Admissibility of estimators
Chapter 8: General decision theory
41. Experiments and their L-spaces
42. Decision functions
43. Lower semicontinuity
44. Risk functions
45. A general minimax theorem
46. The minimax theorem of decision theory
47. Bayes solutions and the complete class theorem
48. The generalized theorem of Hunt and Stein
Chapter 9: Comparison of experiments
49. Basic concepts
50. Standard decision problems
51. Comparison of experiments by standard decision problems
52. Concave function criteria
53. Hellinger transforms and standard measures
54. Comparison of experiments by testing problems
55. The randomization criterion
56. Conical measures
57. Representation of experiments
58. Transformation groups and invariance
59. Topological spaces of experiments
Chapter 10: Asymptotic decision theory
60. Weakly convergent sequences of experiments
61. Contiguous sequences of experiments
62. Convergence in distribution of decision functions
63. Stochastic convergence of decision functions
64. Convergence of minimum estimates
65. Uniformly integrable experiments
66. Uniform tightness of generalized Bayes estimates
67. Convergence of generalized Bayes estimates
Chapter 11: Gaussian shifts on Hilbert spaces
68. Linear stochastic processes and cylinder set measures
69. Gaussian shift experiments
70. Banach sample spaces
71. Testing for Gaussian shifts
72. Estimation for Gaussian shifts
73. Testing and estimation for Banach sample spaces
Chapter 12: Differentiability and asymptotic expansions
74. Stochastic expansion of likelihood ratios
75. Differentiable curves
76. Differentiable experiments
77. Conditions for differentiability
78. Examples of differentiable experiments
79. The stochastic expansion of a differentiable experiment
Chapter 13: Asymptotic normality
80. Asymptotic normality
81. Exponential approximation and asymptotic sufficiency
82. Application to testing hypotheses
83. Application to estimation
84. Characterization of central sequences
Appendix: Notation and terminology
References
List of symbols
Author index
Subject index