Synopsis
0. Introduction.- 0.1. Outline of contents.- 0.2. A few preliminaries.- 1. Likelihood and auxiliary statistics.- 1.1. Likelihood.- 1.2. Moments and cumulants of log likelihood derivatives.- 1.3. Parametrization invariance.- 1.4. Marginal and conditional likelihood.- 1.5. Combinants, auxiliaries, and the p*-model.- 1.6. Orthogonal parameters.- 1.7. Pseudo likelihood, profile likelihood and modified profile likelihood.- 1.8. Ancillarity and conditionality.- 1.9. Partial sufficiency and partial ancillarity.- 1.10. Likelihood expansions.- 1.11. Additional bibliographical notes.- 2. Transformation models and exponential models.- 2.1. Group actions and invariant measures.- 2.2. Transformation models.- 2.3. Transformation submodels.- 2.4. Exponential models.- 2.5. Exponential transformation models.- 2.6. Additional bibliographical notes.- 3. Reparametrizations and differential geometry.- 3.1. Multiarrays.- 3.2. Tensors and affine connections.- 3.3. Strings.- 3.4. Covariant differentiation and strings.- 3.5. Intertwining.- 3.6. Submanifolds.- 3.7. Geometric measures.- 3.8. Manifolds with a Lie group action.- 3.9. Fibre bundles, connections and (parallel) transport.- 3.10. Additional bibliographical notes.- 4. Inferential and geometric structures.- 4.1. Ancillary statistics and conditionality structures.- 4.2. Conditionality structures for transformation models.- 4.3. Construction of approximately ancillary statistics.- 4.4. Jacobians of conditionality structures.- 4.5. Geometry of parametric models.- 4.6. Additional bibliographical notes.- 5. Cumulants.- 5.1. Elemental properties of cumulants.- 5.2. Relations between moments and cumulants.- 5.3. An alternative definition of generalized cumulants.- 5.4. Additional bibliographical notes.- 6. Laplace's method. Edgeworth and saddle-point approximations.- 6.1. Laplace's method.- 6.2. Hermite polynomials.- 6.3. Edgeworth approximations.- 6.4. Saddle-point approximations.- 6.5. Additional bibliographical notes.- 7. Distributions of likelihood quantities.- 7.1. The distribution of the maximum likelihood estimator.- 7.2. Expansion of p*.- 7.3. The distribution of the score vector.- 7.4. The distribution of likelihood ratio statistics.- 7.5. Modified profile likelihood.- 7.6. Additional bibliographical notes.- Appendices.- A.1. Taylor's formula.- A.2. Fourier transformation.- A.3. Some formulas for matrices and determinants.- A.4. Partially ordered sets, partitions and Möbius inversion.- A.5. The Legendre transform.- A.6. A differential geometric inversion result.- References.
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