Contributions to a General Asymptotic Statistical Theory
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.
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The local structure of families of probability measures
Examples of tangent spaces
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According alternatives applied approximable arbitrary assertion Assume assumptions asymptotic asymptotically efficient belongs canonical gradient Chapter close concentration consider contains continuous converges convex corollary corresponding defined definition denote density dependence derivative determined differentiable direction distance distribution efficient efficient estimators envelope power function equality equivalent estimator-sequence Example exists expectation fact family of p-measures fixed given Hence holds hypothesis implies independent instance known leads Lebesgue density Lemma linear mean median natural neighborhood normal observations obtain optimal orthogonal p-measures parametric parametric family particular path positive possible probability measures problem procedure projection Proof properties Proposition prove regularity conditions relation remains Remark requires respect restriction sample sense sequence Statist subfamily sufficiently suggest symmetric tangent cone tangent space term tests Theorem theory tion uniform uniformly variance bound zero