The aso theory developed in Chapters 8 - 12 presumes that the tan gent cones are linear spaces. In the present chapter we collect a few natural examples where the tangent cone fails to be a linear space. These examples are to remind the reader that an extension of the theo ry to convex tangent cones is wanted. Since the results are not needed in the rest of the book, we are more generous ab out regularity condi tions. The common feature of the examples is the following: Given a pre order (i.e., a reflexive and transitive order relation) on a family of p-measures, and a subfamily i of order equivalent p-measures, the fa mily ~ consists of p-measures comparable with the elements of i. This usually leads to a (convex) tangent cone 1f only p-measures larger (or smaller) than those in i are considered, or to a tangent co ne con sisting of a convex cone and its reflexion about 0 if both smaller and larger p-measures are allowed. For partial orders (i.e., antisymmetric pre-orders), ireduces to a single p-measure. we do not assume the p-measures in ~ to be pairwise comparable.

Inhalt
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. 'Full' 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 projections ofdensities,.- 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 criticalregion,.- 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 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.