Asymptotic Methods in Statistical Decision Theory

1. Front Matter

   Pages i-xxvi

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2. Experiments—Decision Spaces

   • Lucien Le Cam

   Pages 1-15

3. Some Results from Decision Theory: Deficiencies

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   Pages 16-28

4. Likelihood Ratios and Conical Measures

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   Pages 29-45

5. Some Basic Inequalities

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   Pages 46-56

6. Sufficiency and Insufficiency

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   Pages 57-80

7. Domination, Compactness, Contiguity

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   Pages 81-95

8. Some Limit Theorems

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   Pages 96-117

9. Invariance Properties

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   Pages 118-153

10. Infinitely Divisible, Gaussian, and Poisson Experiments

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    Pages 154-171

11. Asymptotically Gaussian Experiments: Local Theory

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    Pages 172-205

12. Asymptotic Normality—Global

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    Pages 206-323

13. Posterior Distributions and Bayes Solutions

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    Pages 324-345

14. An Approximation Theorem for Certain Sequential Experiments

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    Pages 346-369

15. Approximation by Exponential Families

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    Pages 370-398

16. Sums of Independent Random Variables

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    Pages 399-456

17. Independent Observations

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    Pages 457-554

18. Independent Identically Distributed Observations

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    Pages 555-633

19. Back Matter

    Pages 634-742

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This book grew out of lectures delivered at the University of California, Berkeley, over many years. The subject is a part of asymptotics in statistics, organized around a few central ideas. The presentation proceeds from the general to the particular since this seemed the best way to emphasize the basic concepts. The reader is expected to have been exposed to statistical thinking and methodology, as expounded for instance in the book by H. Cramer [1946] or the more recent text by P. Bickel and K. Doksum [1977]. Another pos­ sibility, closer to the present in spirit, is Ferguson [1967]. Otherwise the reader is expected to possess some mathematical maturity, but not really a great deal of detailed mathematical knowledge. Very few mathematical objects are used; their assumed properties are simple; the results are almost always immediate consequences of the definitions. Some objects, such as vector lattices, may not have been included in the standard background of a student of statistics. For these we have provided a summary of relevant facts in the Appendix. The basic structures in the whole affair are systems that Blackwell called "experiments" and "transitions" between them. An "experiment" is a mathe­ matical abstraction intended to describe the basic features of an observational process if that process is contemplated in advance of its implementation. Typically, an experiment consists of a set E> of theories about what may happen in the observational process.

• Likelihood
• Random variable
• Variance
• decision theory
• statistics

• Department of Statistics, University of California, Berkeley, USA

  Lucien Le Cam

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