UID:
almahu_9949385564402882
Umfang:
1 online resource (472 pages)
Ausgabe:
First edition.
ISBN:
9781003049340
,
1003049346
,
9781000488012
,
1000488012
,
9781000488074
,
1000488071
Serie:
Chapman & Hall texts in statistical science
Inhalt:
"The purpose of applying mathematical theory to the theory of statistical inference is to make it simpler and more elegant. Theory of Statistical Inference is concerned with the development of a type of optimization theory which can be used to inform the choice of statistical methodology. Of course, this would be pointless without reference to such methods. We are simply noting that they are included in support of the larger goal. This book distinguishes itself from other graduate textbooks because it is written from the point of view that some degree of understanding of measure theory, as well as other branches of mathematics, which include topology, group theory and complex analysis, should be a part of the canon of statistical inference"--
Anmerkung:
Preface 1 Distribution Theory 1.1 Introduction1.2 Probability Measures1.3 Some Important Theorems of Probability1.4 Commonly Used Distributions1.5 Stochastic Order Relations1.6 Quantiles1.7 Inversion of the CDF1.8 Transformations of Random Variables1.9 Moment Generating Functions1.10 Moments and Cumulants1.11 Problems2 Multivariate Distributions2.1 Introduction2.2 Parametric Classes of Multivariate Distributions2.3 Multivariate Transformations2.4 Order Statistics2.5 Quadratic Forms, Idempotent Matrices and Cochran's Theorem2.6 MGF and CGF of Independent Sums2.7 Multivariate Extensions of the MGF2.8 Problems3 Statistical Models3.1 Introduction3.2 Parametric Families for Statistical Inference3.3 Location-Scale Parameter Models3.4 Regular Families3.5 Fisher Information3.6 Exponential Families3.7 Sufficiency3.8 Complete and Ancillary Statistics3.9 Conditional Models and Contingency Tables3.10 Bayesian Models3.11 Indifference, Invariance and Bayesian Prior Distributions3.12 Nuisance Parameters3.13 Principles of Inference3.14 Problems4 Methods of Estimation4.1 Introduction4.2 Unbiased Estimators4.3 Method of Moments Estimators4.4 Sample Quantiles and Percentiles4.5 Maximum Likelihood Estimation4.6 Confidence Sets4.7 Equivariant Versus Shrinkage Estimation4.8 Bayesian Estimation4.9 Problems5 Hypothesis Testing5.1 Introduction5.2 Basic Definitions5.3 Principles of Hypothesis Tests5.4 The Observed Level of Significance (P-Values)5.5 One and Two Sided Tests5.6 Hypothesis Tests and Pivots5.7 Likelihood Ratio Tests5.8 Similar Tests5.9 Problems6 Linear Models6.1 Introduction6.2 Linear Models - Definition6.3 Best Linear Unbiased Estimators (BLUE)6.4 Least-squares Estimators, BLUEs and Projection Matrices6.5 Ordinary and Generalized Least-Squares Estimators6.6 ANOVA Decomposition and the F Test for Linear Models6.7 The F Test for One-Way ANOVA6.8 Simultaneous Confidence Intervals6.9 Multiple Linear Regression6.10 Problems7 Decision Theory7.1 Introduction7.2 Ranking Estimators by MSE7.3 Prediction7.4 The Structure of Decision Theoretic Inference7.5 Loss and Risk7.6 Uniformly Minimum Risk Estimators (The Location-Scale Model7.7 Some Principles of Admissibility7.8 Admissibility for Exponential Families (Karlin's Theorem)7.9 Bayes Decision Rules7.10 Admissibility and Optimality7.11 Problems8 Uniformly Minimum Variance Unbiased (UMVU) Estimation8.1 Introduction8.2 Definition of UMVUE's8.3 UMVUE's and Sufficiency8.4 Methods of Deriving UMVUEs8.5 Nonparametric Estimation and U-statistics8.6 Rank Based Measures of Correlation8.7 Problems9 Group Structure and Invariant Inference9.1 Introduction9.2 MRE Estimators for Location Parameters9.3 MRE Estimators for Scale Parameters9.4 Invariant Density Families9.5 Some Applications of Invariance9.6 Invariant Hypothesis Tests9.7 Problems10 The Neyman-Pearson Lemma10.1 Introduction10.2 Hypothesis Test as Decision Rules10.3 Neyman-Pearson (NP) Tests10.4 Monotone Likelihood Ratios (MLR)10.5 The Generalized Neyman-Pearson Lemma10.6 Invariant Hypothesis Tests10.7 Permutation Invariant Tests10.8 Problems11 Limit Theorems11.1 Introduction11.2 Limits of Sequences of Random Variables11.3 Limits of Expected Values11.4 Uniform Integrability11.5 The Law of Large Numbers11.6 Weak Convergence11.7 Multivariate Extensions of Limit Theorems11.8 The Continuous Mapping Theorem11.9 MGFs, CGFs and Weak Convergence11.10 The Central Limit Theorem for Triangular Arrays11.11 Weak Convergence of Random Vectors11.12 Problems12 Large Sample Estimation - Basic Principles12.1 Introduction12.2 The _-Method12.3 Variance Stabilizing Transformations12.4 The _-Method and Higher Order Approximations12.5 The Multivariate _-Method12.6 Approximating the Distributions of Sample Quantiles: The Bahadur Representation Theorem12.7 A Central Limit Theorem for U-statistics12.8 The Information Inequality12.9 Asymptotic Efficiency12.10 Problems13 Asymptotic Theory for Estimating Equations13.1 Introduction13.2 Consistency and Asymptotic Normality of M-estimators13.3 Asymptotic Theory of MLEs13.4 A General Form for Regression Models13.5 Nonlinear Regression13.6 Generalized Linear Models (GLMs)13.7 Generalized Estimating Equations (GEE)13.8 Consistency of M-estimators13.9 Asymptotic Distribution of ˆ_n 13.10 Regularity Conditions for Estimating Equations13.11 Problems14 Large Sample Hypothesis Testing14.1 Introduction14.2 Model Assumptions14.3 Large Sample Tests for Simple Hypotheses14.4 Nuisance Parameters and Composite Null Hypotheses14.5 A Comparison of the LR, Wald and Score Tests14.6 Pearson's _2 Test for Independence in Contingency Tables14.7 Estimating Power for Approximate _2 Tests14.8 ProblemsA Parametric Classes of DensitiesB Topics in Linear AlgebraB.1 NumbersB.2 Equivalence RelationsB.3 Vector SpacesB.4 MatricesB.5 Dimension of a Subset of Rd C Topics in Real Analysis and Measure TheoryC.1 Metric spacesC.2 Measure TheoryC.3 IntegrationC.4 Exchange of Integration and DifferentiationC.5 The Gamma and Beta FunctionsC.6 Stirling's Approximation of the FactorialC.7 The Gradient Vector and the Hessian MatrixC.8 Normed Vector SpacesC.9 Taylor's TheoremD Group TheoryD.1 Definition of a GroupD.2 SubgroupsD.3 Group HomomorphismsD.4 Transformation GroupsD.5 Orbits and Maximal InvariantsBibliographyIndex
Sprache:
Englisch
Schlagwort(e):
Electronic books.
URL:
https://www.taylorfrancis.com/books/9781003049340
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