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.
Contributions to a General Asymptotic Statistical Theory
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Description
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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