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AuthorPfanzagl, J. author
TitleContributions to a General Asymptotic Statistical Theory [electronic resource] / by J. Pfanzagl
ImprintNew York, NY : Springer New York, 1982
Connect tohttp://dx.doi.org/10.1007/978-1-4612-5769-1
Descript 315 p. online resource

CONTENT

0. Introduction -- 0.1. Why asymptotic theory? -- 0.2. The object of a unified asymptotic theory, -- 0.3. Models, -- 0.4. Functionals, -- 0.5. What are the purposes of this book? -- 0.6. A guide to the contents, -- 0.7. Adaptiveness, -- 0.8. Robustness, -- 0.9. Notations, -- 1. The local structure of families of probability measures -- 1.1. The tangent cone T(P,?), -- 1.2. Properties of T(P,?) - properties of ?, -- 1.3. Convexity of T(P,?), -- 1.4. Symmetry of T(P,?), -- 1.5. Tangent spaces of induced measures, -- 2. Examples of tangent spaces -- 2.1. โ{128}{152}Fullโ{128}{153} tangent spaces, -- 2.2. Parametric families, -- 2.3. Families of symmetric distributions, -- 2.4. Measures on product spaces, -- 2.5. Random nuisance parameters, -- 2.6. A general model, -- 3. Tangent cones -- 3.1. Introduction, -- 3.2. Order with respect to location, -- 3.3. Order with respect to concentration, -- 3.4. Order with respect to asymmetry, -- 3.5. Monotone failure rates, -- 3.6. Positive dependence, -- 4. Differentiable functionals -- 4.1. The gradient of a functional, -- 4.2. Projections into convex sets, -- 4.3. The canonical gradient, -- 4.4. Multidimensional functionals, -- 4.5. Tangent spaces and gradients under side conditions, -- 4.6. Historical remarks, -- 5. Examples of differentiable functionals -- 5.1. Von Mises functionals, -- 5.2. Minimum contrast functionals, -- 5.3. Parameters, -- 5.4. Quantiles, -- 5.5. A location functional, -- 6. Distance functions for probability measures -- 6.1. Some distance functions, -- 6.2. Asymptotic relations between distance functions, -- 6.3. Distances in parametric families, -- 6.4. Distances for product measures, -- 7. Projections of probability measures -- 7.1. Motivation, -- 7.2. The projection, -- 7.3. Projections defined by distances, -- 7.4. Projections of measures โ{128}{148} projections of densities, -- 7.5. Iterated projections, -- 7.6. Projections into a parametric family, -- 7.7. Projections into a family of product measures, -- 7.8. Projections into a family of symmetric distributions, -- 8. Asymptotic bounds for the power of tests -- 8.1. Hypotheses and co-spaces, -- 8.2. The dimension of the co-space, -- 8.3. The concept of asymptotic power functions, -- 8.4. The asymptotic envelope power function, -- 8.5. The power function of asymptotically efficient tests, -- 8.6. Restrictions of the basic family, -- 8.7. Asymptotic envelope power functions using the Hellinger distance, -- 9. Asymptotic bounds for the concentration of estimators -- 9.1. Comparison of concentrations, -- 9.2. Bounds for asymptotically median unbiased estimators, -- 9.3. Multidimensional functionals, -- 9.4. Locally uniform convergence, -- 9.5. Restrictions of the basic family, -- 9.6. Functionals of induced measures, -- 10. Existence of asymptotically efficient estimators for probability measures -- 10.1. Asymptotic efficiency, -- 10.2. Density estimators, -- 10.3. Parametric families, -- 10.4. Projections of estimators, -- 10.5. Projections into a parametric family, -- 10.6. Projections into a family of product measures, -- 11. Existence of asymptotically efficient estimators for functionals -- 11.1. Introduction, -- 11.2. Asymptotically efficient estimators for functionals from asymptotically efficient estimators for probability measures, -- 11.3. Functions of asymptotically efficient estimators are asymptotically efficient, -- 11.4. Improvement of asymptotically inefficient estimators, -- 11.5. A heuristic justification of the improvement procedure, -- 11.6. Estimators with stochastic expansion, -- 12. Existence of asymptotically efficient tests -- 12.1. Introduction, -- 12.2. An asymptotically efficient critical region, -- 12.3. Hypotheses on functionals, -- 13. Inference for parametric families -- 13.1. Estimating a functional, -- 13.2. Variance bounds for parametric subfamilies, -- 13.3. Asymptotically efficient estimators for parametric subfamilies, -- 14. Random nuisance parameters -- 14.1. Introduction, -- 14.2. Estimating a structural parameter in the presence of a known random nuisance parameter, -- 14.3. Estimating a structural parameter in the presence of an unknown random nuisance parameter, -- 15. Inference for symmetric probability measures -- 15.1. Asymptotic variance bounds for functionals of symmetric distributions, -- 15.2. Asymptotically efficient estimators for functionals of symmetric distributions, -- 15.3. Symmetry in two-dimensional distributions, -- 16. Inference for measures on product spaces -- 16.1. Introduction, -- 16.2. Variance bounds, -- 16.3. Asymptotically efficient estimators for product measures, -- 16.4. Estimators for von Mises functionals, -- 16.5. A special example, -- 17. Dependence โ{128}{148} independence -- 17.1. Measures of dependence, -- 17.2. Estimating measures of dependence, -- 17.3. Tests for independence, -- 18. Two-sample problems -- 18.1. Introduction, -- 18.2. Inherent relationships between x and y, -- 18.3. The tangent spaces, -- 18.4. Testing for equality, -- 18.5. Estimation of a transformation parameter, -- 18.6. Estimation in the proportional failure rate model, -- 18.7. Dependent samples, -- 19. Appendix -- 19.1. Miscellaneous lemmas, -- 19.2. Asymptotic normality of log-likelihood ratios, -- References -- Notation index -- Author index


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