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AuthorHeyer, H. author
TitleTheory of Statistical Experiments [electronic resource] / by H. Heyer
ImprintNew York, NY : Springer New York, 1982
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Descript X, 289 p. online resource


By a statistical experiment we mean the procedure of drawing a sample with the intention of making a decision. The sample values are to be regarded as the values of a random variable defined on some measยญ urable space, and the decisions made are to be functions of this random variable. Although the roots of this notion of statistical experiment extend back nearly two hundred years, the formal treatment, which involves a description of the possible decision procedures and a conscious attempt to control errors, is of much more recent origin. Building upon the work of R. A. Fisher, J. Neyman and E. S. Pearson formalized many deciยญ sion problems associated with the testing of hypotheses. Later A. Wald gave the first completely general formulation of the problem of statistiยญ cal experimentation and the associated decision theory. These achieveยญ ments rested upon the fortunate fact that the foundations of probability had by then been laid bare, for it appears to be necessary that any such quantitative theory of statistics be based upon probability theory. The present state of this theory has benefited greatly from contriยญ butions by D. Blackwell and L. LeCam whose fundamental articles expanded the mathematical theory of statistical experiments into the field of comยญ parison of experiments. This will be the main motivation for the apยญ proach to the subject taken in this book


I. Games and Statistical Decisions -- ยง 1. Two-Person Zero Sum Games -- ยง 2. Concave-Convex Games and Optimality -- ยง 3. Basic Principles of Statistical Decision Theory -- II. Sufficient ?-Algebras and Statistics -- ยง 4. Generalities -- ยง 5. Properties of the System of All Sufficient ?-Algebras -- ยง 6. Completeness and Minimal Sufficiency -- III. Sufficiency under Additional Assumptions -- ยง 7. Sufficiency in the Separable Case -- ยง 8. Sufficiency in the Dominated Case -- ยง 9. Examples and Counter-Examples -- IV. Testing Experiments -- ยง10. Fundamentals -- ยง11. Construction of Most Powerful Tests -- ยง12. Least Favorable Distributions and Bayes Tests -- V. Testing Experiments Admitting an Isotone Likelihood Quotient -- ยง13. Isotone Likelihood Quotient -- ยง14. One-Dimensional Exponential Experiments -- ยง15. Similarity, Stringency and Unbiasedness -- VI. Estimation Experiments -- ยง16. Minimum Variance Unbiased Estimators -- ยง17. p-Minimality -- ยง18. Estimation Via the Order Statistic -- VII. Information and Sufficiency -- ยง19. Comparison of Classical Experiments -- ยง20. Representation of Positive Linear Operators by Stochastic Kernels -- ยง21. The Stochastic Kernel Criterion -- ยง22. Sufficiency in the Sense of Blackwell -- VIII. Invariance and the Comparison of Experiments -- ยง23. Existence of Invariant Stochastic Kernels -- ยง24. Comparison of Translation Experiments -- ยง25. Comparison of Linear Normal Experiments -- IX. Comparison of Finite Experiments -- ยง26. Comparison by k-Decision Problems -- ยง27. Comparison by Testing Problems -- ยง28. Standard Experiments -- ยง29. General Theory of Standard Measures -- ยง30. Sufficiency and Completeness -- X. Comparison with Extremely Informative Experiments -- ยง31. Bayesian Deficiency -- ยง32. Totally Informative Experiments -- ยง33. Totally Uninformative Experiments -- ยง34. Inequalities Between Deficiencies -- Notational Conventions -- References -- Symbol Index

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