Statistical Decision Rules and Optimal InferenceNone available in plain English. |
Contents
| 1 | |
| 13 | |
3 Smooth manifolds and their mappings | 37 |
4Category theory and geometry | 50 |
THE FORMAL DECISION PROBLEM | 65 |
6 Markov geometry of families of probability distributions | 76 |
7 Geometry of dominated families of probability distributions | 92 |
8 Invariant information characteristics | 113 |
16 Continuously differentiable families of probability distributions | 228 |
GEOMETRY OF EXPONENT FAMILIES OF PROBABILITY DISTRIBUTIONS In this chapter we develop the theory of exponent families of prob... | 245 |
18 Exponent families of probability distributions | 262 |
19 Natural parametrization of exponent families | 279 |
20 Conjugate parametrizations of an exponent family of probability distributions | 290 |
21 Distributions of values of the directional statistic of an exponent family and related families | 303 |
22 Nonsynunetric Pythagorean geometry of the information deviation | 319 |
23 Charts of an exponent family of probability distributions | 334 |
EQUIVARIANT DIFFERENTIAL GEOMETRY OF A COLLECTION OF PROBABILITY DISTRIBUTIONS | 127 |
10 Project ive geometry of a collection of probability distributions | 140 |
11 Invariant Riemannian metric on a manifold of probability distributions | 156 |
12 Geodesic mean of probability distributions | 165 |
SMOOTH FAMILIES OF PROBABILITY DISTRIBUTIONS AND THE INFORMATION INEQUALITY 13 Finitedimensional approximation of infi... | 185 |
14 Differentiate families of probability distributions | 199 |
15 The information inequality | 213 |
OPTIMAL DECISION RULES IN THE EQUIVARIANT PROBLEM OF POINT ESTIMATION | 355 |
25 Estimation of the unknown density of a probability distribution of observations | 370 |
26 Invariant loss functions in problems of mathematical statistics 1 In the last two sections we considered wellknown examples of statistical point esti... | 391 |
27 Optimal estimators for smooth families of probability distributions | 403 |
28 Quasihomogeneous families of probability distributions | 437 |
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A₁ affine algebra atoms barycentric coordinates C₁ canonical parametrization chart conditional distribution congruent consider constructive continuous convex function convex set coordinates COROLLARY corresponding countable covariant decision rule defined definition density differentiable directional statistic domain equal equivalent equivariant estimator exists exponent family families of probability Fin f finite finite-dimensional follows formula geodesic family geodesic mean geometry Hence homomorphism I[PR information deviation integral interval invariant laws Lebesgue Lemma lemma is proved likelihood function ln p(w loss function manifold Markov morphism matrix measurable space metric monotone natural parametrization nonnegative norm normalizing divisor o-algebra obtain one-to-one P{dw P₁ parameter partial derivatives partition probability distributions probability measure random S-measurable S₁ smooth strictly convex subfamily subset sufficient statistic tangent vectors tensor Theorem tion transformation unique values vector field X₁ zero Πω σα
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