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课程： 计算统计（Computational Statistics）
授课教师： 陈 颖 副教授
课程类别： 专业核心课、实验课程
教材和参考书目：
指定教材：Sheldon M. Ross, Simulation, 4th Edition, 人民邮电出版社影印版，2007。
参考教材：
Wendy L. Martinez, Angel R. Martinez, Computational Statistics Handbook with Matlab, Chapman.
茆诗松、王静龙、濮晓龙，高等数理统计，高等教育出版社，1998。（第六章）
教学目的
本课程介绍统计模拟的一些实用方法和技术。我们将对概率的基本知识进行了简单的回顾，然后介绍如何利用计算机产生随机数以及如何利用这些随机数产生任意分布的随机变量、随机过程等，并介绍一些分析统计数据的方法和技术，如Bootstrap、方差缩减技术等。将介绍如何利用统计模拟来判断所选的随机模型是否拟合实际的数据。最后介绍MCMC及一些最新发展的统计模拟技术和论题。我们将采用Matlab软件实现所涉及的算法程序。
计算统计教学要点
Elements of Probabilily
(Chapter 2)
Sample Space and Events
Axioms of Probability
Conditional Probability and Independence
Random Variables
Expectation
Variance
Chebyshev’s Inequality and the Laws of Large Numbers
Some Discrete Random Variables
Binomial Random Variables
Poisson Random Variables
Geometric Random Variables
The Negative Binomial Random Variable
Hypergeometric Random Variables
Continuous Random Variables
Uniformly Distributed Random Variables
Normal Random Variables
Exponential Random Variables
The Poisson Process and Gamma Random Variables
The Nonhomogeneous Poisson Process
Conditional Expectation and Conditional Variance
Random Numbers (Chapter 3)
Pseudorandom Number Generation
Using Random Numbers to Evaluate Integrals
Generating Discrete Random Variables (Chapter 4)
The Inverse Transform Method
Generating a Poisson Random Variable
Generating Binomial Random Variables
The Acceptance-Rejection Technique
The Composition Approach
Generating Random Vectors
Generating Continuous Random Variables
The Inverse Transform Algorithm
The Rejection Method
The Polar Method for Generating Normal Random Variables
Generating a Poisson Process
Generating a Nonhomogeneous Poisson Process
Statistical Analysis of Simulated Data (Chapter 7)
The Sample Mean and Sample Variance
Interval Estimates of a Population Mean
The Bootstrapping Technique for Estimating Mean Square Errors
Variance Reduction Techniques (Chapter 8)
The Use of Antithetic Variables
The Use of Control Variates
Variance Reduction by Conditioning
Estimating the Expected Number of Renewals by Time t
Stratified Sampling
Importance Sampling
Using Common Random Numbers
Evaluating an Exotic Option
Statistical Validation Techniques (Chapter 9)
Goodness of Fit Tests
The Chi-Square Goodness of Fit Test for Discrete Data
The Kolmogorov-Smirnov Test for Continuous Data
Goodness of Fit Tests When Some Parameters Are Unspecified
The Discrete Data Case
The Continuous Data Case
The Two-Sample Problem
Validating the Assumption of a Nonhomogeneous Poisson Process
Markov Chain Monte Carlo Methods （Chapter 10）
Markov Chains
The Hastings-Metropolis Algorithm
The Gibbs Sampler
Simulated Annealing
The Sampling Importance Resampling Algorithm
Additional Topics
EM Algorithm
Jacknife
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