Theory of Statistical Experiments

ISBN-13: 9781461382201 / Angielski / Miękka / 2011 / 289 str.

H. Heyer

Opis

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 meas 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 deci sion problems associated with the testing of hypotheses. Later A. Wald gave the first completely general formulation of the problem of statisti cal experimentation and the associated decision theory. These achieve 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 contri butions by D. Blackwell and L. LeCam whose fundamental articles expanded the mathematical theory of statistical experiments into the field of com parison of experiments. This will be the main motivation for the ap proach to the subject taken in this book."

Spis treści

I. Games and Statistical Decisions.- § 1. Two-Person Zero Sum Games.- § 2. Concave-Convex Games and Optimality.- § 3. Basic Principles of Statistical Decision Theory.- II. Sufficient ?-Algebras and Statistics.- § 4. Generalities.- § 5. Properties of the System of All Sufficient ?-Algebras.- § 6. Completeness and Minimal Sufficiency.- III. Sufficiency under Additional Assumptions.- § 7. Sufficiency in the Separable Case.- § 8. Sufficiency in the Dominated Case.- § 9. Examples and Counter-Examples.- IV. Testing Experiments.- §10. Fundamentals.- §11. Construction of Most Powerful Tests.- §12. Least Favorable Distributions and Bayes Tests.- V. Testing Experiments Admitting an Isotone Likelihood Quotient.- §13. Isotone Likelihood Quotient.- §14. One-Dimensional Exponential Experiments.- §15. Similarity, Stringency and Unbiasedness.- VI. Estimation Experiments.- §16. Minimum Variance Unbiased Estimators.- §17. p-Minimality.- §18. Estimation Via the Order Statistic.- VII. Information and Sufficiency.- §19. Comparison of Classical Experiments.- §20. Representation of Positive Linear Operators by Stochastic Kernels.- §21. The Stochastic Kernel Criterion.- §22. Sufficiency in the Sense of Blackwell.- VIII. Invariance and the Comparison of Experiments.- §23. Existence of Invariant Stochastic Kernels.- §24. Comparison of Translation Experiments.- §25. Comparison of Linear Normal Experiments.- IX. Comparison of Finite Experiments.- §26. Comparison by k-Decision Problems.- §27. Comparison by Testing Problems.- §28. Standard Experiments.- §29. General Theory of Standard Measures.- §30. Sufficiency and Completeness.- X. Comparison with Extremely Informative Experiments.- §31. Bayesian Deficiency.- §32. Totally Informative Experiments.- §33. Totally Uninformative Experiments.- §34. Inequalities Between Deficiencies.- Notational Conventions.- References.- Symbol Index.