统计模拟

1 Introduction

Exercises

2 Elements of Probability

2.1 Sample Space and Events

2.2 Axioms of Probability

2.3 Conditional Probability and Independence

2.4 Random Variables

2.5 Expectation

2.6 Variance

2.7 Chebyshevs Inequality and the Laws of Large Numbers

2.8 Some Discrete Random Variables

Binomial Random Variables

Poisson Random Variables

Geometric Random Variables

The Negative Binomial Random Variable

Hypergeometric Random Variables

2.9 Continuous Random Variables

Uniformly Distributed Random Variables

Normal Random Variables

Exponential Random Variables

The Poisson Process and Gamma Random Variables

The Nonhomogeneous Poisson Process

2.10 Conditional Expectation and Conditional Variance

Exercises

References

3 Random Numbers

Introduction

3.1 Pseudorandom Number Generation

3.2 Using Random Numbers to Evaluate Integrals

Exercises

References

4 Generating Discrete Random Variables

4.1 The Inverse Transform Method

4.2 Generating a Poisson Random Variable

4.3 Generating Binomial Random Variables

4.4 The Acceptance-Rejection Technique

4.5 The Composition Approach

4.6 Generating Random Vectors

Exercises

5 Generating Continuous Random Variables

Introduction

5.1 The Inverse Transform Algorithm

5.2 The Rejection Method

5.3 The Polar Method for Generating Normal Random Variables

5.4 Generating a Poisson Process

5.5 Generating a Nonhomogeneous Poisson Process

Exercises

References

6 The Discrete Event Simulation Approach

Introduction

6.1 Simulation via Discrete Events

6.2 A Single-Server Queueing System

6.3 A Queueing System with Two Servers in Series

6.4 A Queueing System with Two Parallel Servers

6.5 An Inventory Model

6.6 An Insurance Risk Model

6.7 A Repair Problem

6.8 Exercising a Stock Option

6.9 Verification of the Simulation Model

Exercises

References

7 Statistical Analysis of Simulated Data

Introduction

7.1 The Sample Mean and Sample Variance

7.2 Interval Estimates of a Population Mean

7.3 The Bootstrapping Technique for Estimating Mean Square Errors

Exercises

References

8 Variance Reduction Techniques

Introduction

8.1 The Use of Antithetic Variables

8.2 The Use of Control Variates

8.3 Variance Reduction by Conditioning

Estimating the Expected Number of Renewals by Time t

8.4 Stratified Sampling

8.5 Importance Sampling

8.6 Using Common Random Numbers

8.7 Evaluating an Exotic Option

Appendix： Verification of Antithetic Variable Approach

When Estimating the Expected Value of Monotone Functions

Exercises

References

9 Statistical Validation Techniques

Introduction

9.1 Goodness of Fit Tests

The Chi-Square Goodness of Fit Test for Discrete Data

The Kolmogorov-Smirnov Test for Continuous Data

9.2 Goodness of Fit Tests When Some Parameters Are Unspecified

The Discrete Data Case

The Continuous Data Case

9.3 The Two-Sample Problem

9.4 Validating the Assumption of a Nonhomogeneous

Poisson Process

Exercises

References

10 Markov Chain Monte Carlo Methods

Introduction

10.1 Markov Chains

10.2 The Hastings-Metropolis Algorithm

10.3 The Gibbs Sampler

10.4 Simulated Annealing

10.5 The Sampling Importance Resampling Algorithm

Exercises

References

11 Some Additional Topics

Introduction

11.1 The Alias Method for Generating Discrete Random Variables

11.2 Simulating a Two-Dimensional Poisson Process

11.3 Simulation Applications of an Identity for Sums of Bernoulli Random Variables

11.4 Estimating the Distribution and the Mean of the First Passage Time of a Markov Chain

11.5 Coupling from the Past

Exercises

References

Index

“……本书内容丰富，不论作为教材还是参考书都非常值得推荐。”——美国统计学报
　　“本书是一本非常优秀的教材，强调了计算机在模拟技术上的应用。一定的概率和统计知识将有助于理解本书的精髓。”——亚马逊网上书店评论
　　统计模拟是一门新兴的统计学和计算机结合的学科，因其便利性和经济性而广泛应用于统计学、数学、精算科学、工程学、物理学等众多领域，用以获得精确而有效的解决方案。
　　本书是国际知名统计学家Sheldon M. Ross所著的经典教材，已被加州大学伯克利分校、哥伦比亚大学等多所名校采用。书中涵盖了统计模拟最新方法和技术，提供了丰富的实例，备受业界推崇。
　　本书特色：
　　提供了分析模拟数据以及模拟模型的拟合检验所需的统计方法。
　　通过许多实用的例子（如多服务器排队法、存货控制及行使股票期权等）来阐明和提出理论。
　　强调方差缩减技术，包括控制变量及它们在因归分析中的应用等。
　　提供了有关保险风险模型、生成随机向量、奇异期权的材料和关于产生离散随机变量混淆方法的独特材料。
　　第4版特别增加了随机序列函数和随机子集函数的评估、分层抽样法的应用。　
　　本书介绍了统计模拟的一些实用方法和技术。在对概率的基本知识进行了简单的回顾这后，介绍了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。然后介绍一些分析编译数据的方法和技术，如Bootstrap、方差缩减技术等。接着介绍了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。最后介绍了MCMC及一些最新发展的统计模拟技术和论题。本书可作为统计学、计算数学、保险学、精算学等专业本科生教材，也可供相关专业人士参考。本书为英文第4版。