InhaltsangabePreface I Review of Probability and Distribution Theory 1 Probability and Random Variables 1.1 Introduction 1.2 Univariate Discrete Distributions 1.2.1 The Bernoulli and Binomial Distributions 1.2.2 The Poisson Distribution 1.2.3 Binomial Distribution: Normal Approximation 1.3 Univariate Continuous Distributions 1.3.1 The Uniform, Beta, Gamma, Normal, and Student-t Distributions 1.4 Multivariate Probability Distributions 1.4.1 The Multinomial Distribution 1.4.2 The Dirichlet Distribution 1.4.3 The dDimensional Uniform Distribution 1.4.4 The Multivariate Normal Distribution 1.4.5 The Chi-square Distribution 1.4.6 The Wishart and Inverse Wishart Distributions 1.4.7 The Multivariate-t Distribution 1.5 Distributions with Constrained Sample Space 1.6 Iterated Expectations 2 Functions of Random Variables 2.1 Introduction 2.2 Functions of a Single Random Variable 2.2.1 Discrete Random Variables 2.2.2 Continuous Random Variables 2.2.3 Approximating the Mean and Variance 2.2.4 Delta Method 2.3 Functions of Several Random Variables 2.3.1 Linear Transformations 2.3.2 Approximating the Mean and Covariance Matrix II Methods of Inference 3 An Introduction to Likelihood Inference 3.1 Introduction 3.2 The Likelihood Function 3.3 The Maximum Likelihood Estimator 3.4 Likelihood Inference in a Gaussian Model 3.5 Fisher's Information Measure 3.5.1 Single Parameter Case 3.5.2 Alternative Representation of Information 3.5.3 Mean and Variance of the Score Function 3.5.4 Multiparameter Case 3.5.5 CramerRao Lower Bound 3.6 Sufficiency 3.7 Asymptotic Properties: Single Parameter Models 3.7.1 Probability of the Data Given the Parameter 3.7.2 Consistency 3.7.3 Asymptotic Normality and Effciency 3.8 Asymptotic Properties: Multiparameter Models 3.9 Functional Invariance 3.9.1 Illustration of Functional Invariance 3.9.2 Invariance in a Single Parameter Model 3.9.3 Invariance in a Multiparameter Model 4 Further Topics in Likelihood Inference 4.1 Introduction 4.2 Computation of Maximum Likelihood Estimates 4.3 Evaluation of Hypotheses 4.3.1 Likelihood Ratio Tests 4.3.2 Con.dence Regions 4.3.3 Wald's Test 4.3.4 Score Test 4.4 Nuisance Parameters 4.4.1 Loss of Efficiency Due to Nuisance Parameters 4.4.2 Marginal Likelihoods 4.4.3 Profile Likelihoods 4.5 Analysis of a Multinomial Distribution 4.5.1 Amount of Information per Observation 4.6 Analysis of Linear Logistic Models 4.6.1 The Logistic Distribution 4.6.2 Likelihood Function under Bernoulli Sampling 4.6.3 Mixed Effects Linear Logistic Model 5 An Introduction to Bayesian Inference 5.1 Introduction 5.2 Bayes Theorem: Discrete Case 5.3 Bayes Theorem: Continuous Case 5.4 Posterior Distributions 5.5 Bayesian Updating 5.6 Features of Posterior Distributions 5.6.1 Posterior Probabilities 5.6.2 Posterior Quantiles 5.6.3 Posterior Modes 5.6.4 Posterior Mean Vector and Covariance Matrix 6 Bayesian Analysis of Linear Models 6.1 Introduction 6.2 The Linear Regression Model 6.2.1 Inference under Uniform Improper Priors 6.2.2 Inference under Conjugate Priors 6.2.3 Orthogonal Parameterization of the Model 6.3 The Mixed Linear Model 6.3.1 Bayesian View of the Mixed Effects Model 6.3.2 Joint and Conditional Posterior Distributions 6.3.3 Marginal Distribution of Variance Components 6.3.4 Marginal Distribution of Location Parameters 7 The Prior Distribution and Bayesian Analysis 7.1 Introduction 7.2 An Illustration of the Effect of Priors on Inferences 7.3 A Rapid Tour of Bayesian Asymptotics 7.3.1 Discrete Parameter 7.3.2 Continuous Parameter 7.4 Statistical Information and Entropy 7.4.1 Information 7.4.2 Entropy of a Discrete Distribution 7.4.3 Entropy of a Joint and Conditional Distribution 7.4.4 Entropy of a Continuous Distribution 7.4.5 Information about a Parameter 7.4.6 Fisher's Information Revisited 7.