目錄
Contents
Preface
Chapter 1 Generation of Random Variables
1.1 The Inversion Method
1.1.1 Generating samples from continuous distributions
1.1.2 Generating samples from discrete distributions
1.2 The Grid Method
1.3 The Rejection Method
1.3.1 Generating samples from continuous distributions
1.3.2 The efficiency of the rejection method
1.3.3 Several examples
1.3.4 Log-concave densities
1.4 The Sampling/Importance Resampling (SIR) Method
1.4.1 The SIR without replacement
1.4.2 Theoretical justification
1.5 The Stochastic Representation (SR) Method
1.5.1 The『d='operator
1.5.2 Many-to-one SR for univariate case
1.5.3 SR for multivariate case
1.5.4 Mixture representation
1.6 The Conditional Sampling Method
Exercise 1
Chapter 2 Optimization
2.1 A Review of Some Standard Concepts
2.1.1 Order relations
2.1.2 Stationary points
2.1.3 Convex and concave functions
2.1.4 Mean value theorem
2.1.5 Taylor theorem
2.1.6 Rates of convergence
2.1.7 The case of multiple dimensions
2.2 Newton's Method and Its Variants
2.2.1 Newton's method and root finding
2.2.2 Newton's method and optimization
2.2.3 The Newton-Raphson algorithm
2.2.4 The Fisher scoring algorithm
2.2.5 Application to logistic regression
2.3 The Expectation-Maximization (EM) Algorithm
2.3.1 The formulation of the EM algorithm
2.3.2 The ascent property of the EM algorithm
2.3.3 Missing information principle and standard errors
2.4 The ECM Algorithm
2.5 Minorization-Maximization (MM) Algorithms
2.5.1 A brief review of MM algorithms
2.5.2 The MM idea
2.5.3 The quadratic lower-bound algorithm
2.5.4 The De Pierro algorithm
Exercise 2
Chapter 3 Integration
3.1 Laplace Approximations
3.2 Riemannian Simulation
3.2.1 Classical Monte Carlo integration
3.2.2 Motivation for Riemannian simulation
3.2.3 Variance of the Riemannian sum estimator
3.3 The Importance Sampling Method
3.3.1 The formulation of the importance sampling method
3.3.2 The weighted estimator
3.4 Variance Reduction
3.4.1 Antithetic variables
3.4.2 Control variables
Exercise 3
Chapter 4 Markov Chain Monte Carlo Methods
4.1 Bayes Formulae and Inverse Bayes Formulae (IBF)
4.1.1 The point,function- and sampling-wise IBF
4.1.2 Monte Carlo versions of the IBF
4.1.3 Generalization to the case of three random variables
4.2 The Bayesian Methodology
4.2.1 The posterior distribution
4.2.2 Nuisance parameters
4.2.3 Posterior predictive distribution
4.2.4 Bayes factor
4.2.5 Estimation of marginal likelihood
4.3 The Data Augmentation (DA) Algorithm
4.3.1 Missing data mechanism
4.3.2 The idea of data augmentation
4.3.3 The original DA algorithm
4.3.4 Connection with the IBF
4.4 The Gibbs sampler
4.4.1 The formulation of the Gibbs sampling
4.4.2 The two-block Gibbs sampling
4.5 The Exact IBF Sampling
4.6 The IBF sampler
4.6.1 Background and the basic idea
4.6.2 The formulation of the IBF sampler
4.6.3 Theoretical justification for choosing θ0 =.θ
Exercise 4
Chapter 5 Bootstrap Methods
5.1 Bootstrap Confidence Intervals
5.1.1 Parametric bootstrap
5.1.2 Non-parametric bootstrap
5.2 Hypothesis Testing with the Bootstrap
5.2.1 Testing equality of two unknown distributions
5.2.2 Testing equality of two group means
5.2.3 One-sample problem
Exercise 5
Appendix A Some Statistical Distributions and Stochastic Processes
A.1 Discrete Distributions
A.1.1 Finite discrete distribution
A.1.2 Hypergeometric distribution
A.1.3 Binomial and related distributions
A.1.4 Poisson and related distributions
A.1.5 Negative-binomial and related distributions
A.1.6 Generalized Poisson and related distributions
A.1.7 Multinomial and related distributions
A.2 Continuous Distributions
A.2.1 Uniform, beta and Dirichlet distributions
A.2.2 Logistic and Laplace distributions
A.2.3 Exponential, gamma and inverse gamma distributions
A.2.4 Chi-square, F and inverse chi-square distributions
A.2.5 Normal, lognormal and inverse Gaussian distributions
A.2.6 Multivariate normal distribution
A.2.7 Student's t and multivariate t distributions
A.2.8 Wishart and inverse Wishart distributions
A.3 Stochastic Processes
A.3.1 Homogeneous Poisson process
A.3.2 Nonhomogeneous Poisson process
Appendix B R Programming
B.1 Basic Commands
B.1.1 Expressions
B.1.2 Assignment operator
B.2 Vectors and Matrices
B.2.1 Vectors
B.2.2 Matrices
B.3 Lists, Data Frames and Arrays
B.3.1 Lists
B.3.2 Data frames
B.3.3 Arrays
B.4 Flow Control
B.5 User Functions
B.6 Some Commonly-Used R Functions for Data Analysis
Appendix C Introduction of Latent Variables Methods
C.1 MLEs of Parameters in t Distribution
C.2 MLEs of Parameters in the Poisson Additive Model
C.3 MLEs of Parameters in Constrained Normal Models
C.4 Binormal Model with Missing Data
List of Figures
List of Tables
List of Acronyms
List of Symbols
References
Subject Index
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