Mathematical Theory of Statistics
Description
Reviews / Votes
"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
More details
Other editions
Additional editions
Content
- Intro
- Chapter 1: Basic Notions on Probability Measures
- 1. Decomposition of probability measures
- 2. Distances between probability measures
- 3. Topologies and s-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
