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E-raamat: Introduction to Statistical Computing: A Simulation-based Approach

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(School of Mathematics University of Leeds, UK)
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"This is a book about exploring random systems using computer simulation and thus, this book combines two different topic areas which have always fascinated me: the mathematical theory of probability and the art of programming computers"--

Exploring random systems using computer simulations, Voss combines two areas that interest him, the mathematical theory of probability and the art of programming computers. He focuses on three types of questions: what is a typical state of a system, how large the random fluctuations are, and how small the probability is of the system behaving in a specified untypical way. His topics are random number generation, simulating statistical models, Monte Carlo methods, Markov Chain Monte Carlo methods, beyond Monte Carlo, and continuous-time models. Annotation ©2014 Book News, Inc., Portland, OR (booknews.com)

A comprehensive introduction to sampling-based methods in statistical computing

The use of computers in mathematics and statistics has opened up a wide range of techniques for studying otherwise intractable problems. Sampling-based simulation techniques are now an invaluable tool for exploring statistical models. This book gives a comprehensive introduction to the exciting area of sampling-based methods.

An Introduction to Statistical Computing introduces the classical topics of random number generation and Monte Carlo methods. It also includes some advanced methods such as the reversible jump Markov chain Monte Carlo algorithm and modern methods such as approximate Bayesian computation and multilevel Monte Carlo techniques

An Introduction to Statistical Computing:

  • Fully covers the traditional topics of statistical computing.
  • Discusses both practical aspects and the theoretical background.
  • Includes a chapter about continuous-time models.
  • Illustrates all methods using examples and exercises.
  • Provides answers to the exercises (using the statistical computing environment R); the corresponding source code is available online.
  • Includes an introduction to programming in R.

This book is mostly self-contained; the only prerequisites are basic knowledge of probability up to the law of large numbers. Careful presentation and examples make this book accessible to a wide range of students and suitable for self-study or as the basis of a taught course

Arvustused

"The exposition is quite clear, intuitive, and is a useful complement to more abstract treatises on stochastic calculus and simulation."   (MathSciNet, 1 December 2015)

Careful presentation and examples make this book accessible to a wide range of students and suitable for self-study or as the basis of a taught course.  (Zentralblatt MATH, 1 March 2014)

Statistical computing in its broadest sense is an ever-growing field far too extensive to be covered in a single text. The current book has a far more manageable scope, notwithstanding its title. Its focus is on the use of Monte Carlo methods to simulate random systems and explore statistical models.  (Mathematical Association of America, 1 January 2014)

 

List of algorithms ix
Preface xi
Nomenclature xiii
1 Random number generation 1(40)
1.1 Pseudo random number generators
2(6)
1.1.1 The linear congruential generator
2(2)
1.1.2 Quality of pseudo random number generators
4(4)
1.1.3 Pseudo random number generators in practice
8(1)
1.2 Discrete distributions
8(3)
1.3 The inverse transform method
11(4)
1.4 Rejection sampling
15(15)
1.4.1 Basic rejection sampling
15(3)
1.4.2 Envelope rejection sampling
18(4)
1.4.3 Conditional distributions
22(4)
1.4.4 Geometric interpretation
26(4)
1.5 Transformation of random variables
30(6)
1.6 Special-purpose methods
36(1)
1.7 Summary and further reading
36(1)
Exercises
37(4)
2 Simulating statistical models 41(28)
2.1 Multivariate normal distributions
41(4)
2.2 Hierarchical models
45(5)
2.3 Markov chains
50(8)
2.3.1 Discrete state space
51(5)
2.3.2 Continuous state space
56(2)
2.4 Poisson processes
58(9)
2.5 Summary and further reading
67(1)
Exercises
67(2)
3 Monte Carlo methods 69(40)
3.1 Studying models via simulation
69(5)
3.2 Monte Carlo estimates
74(10)
3.2.1 Computing Monte Carlo estimates
75(1)
3.2.2 Monte Carlo error
76(4)
3.2.3 Choice of sample size
80(2)
3.2.4 Refined error bounds
82(2)
3.3 Variance reduction methods
84(12)
3.3.1 Importance sampling
84(4)
3.3.2 Antithetic variables
88(5)
3.3.3 Control variates
93(3)
3.4 Applications to statistical inference
96(10)
3.4.1 Point estimators
97(3)
3.4.2 Confidence intervals
100(3)
3.4.3 Hypothesis tests
103(3)
3.5 Summary and further reading
106(1)
Exercises
106(3)
4 Markov Chain Monte Carlo methods 109(72)
4.1 The Metropolis-Hastings method
110(15)
4.1.1 Continuous state space
110(3)
4.1.2 Discrete state space
113(3)
4.1.3 Random walk Metropolis sampling
116(3)
4.1.4 The independence sampler
119(1)
4.1.5 Metropolis-Hastings with different move types
120(5)
4.2 Convergence of Markov Chain Monte Carlo methods
125(12)
4.2.1 Theoretical results
125(4)
4.2.2 Practical considerations
129(8)
4.3 Applications to Bayesian inference
137(4)
4.4 The Gibbs sampler
141(17)
4.4.1 Description of the method
141(5)
4.4.2 Application to parameter estimation
146(5)
4.4.3 Applications to image processing
151(7)
4.5 Reversible Jump Markov Chain Monte Carlo
158(20)
4.5.1 Description of the method
160(11)
4.5.2 Bayesian inference for mixture distributions
171(7)
4.6 Summary and further reading
178(1)
4.6 Exercises
178(3)
5 Beyond Monte Carlo 181(32)
5.1 Approximate Bayesian Computation
181(11)
5.1.1 Basic Approximate Bayesian Computation
182(6)
5.1.2 Approximate Bayesian Computation with regression
188(4)
5.2 Resampling methods
192(17)
5.2.1 Bootstrap estimates
192(5)
5.2.2 Applications to statistical inference
197(12)
5.3 Summary and further reading
209(1)
Exercises
209(4)
6 Continuous-time models 213(50)
6.1 Time discretisation
213(1)
6.2 Brownian motion
214(7)
6.2.1 Properties
216(1)
6.2.2 Direct simulation
217(1)
6.2.3 Interpolation and Brownian bridges
218(3)
6.3 Geometric Brownian motion
221(3)
6.4 Stochastic differential equations
224(19)
6.4.1 Introduction
224(2)
6.4.2 Stochastic analysis
226(5)
6.4.3 Discretisation schemes
231(5)
6.4.4 Discretisation error
236(7)
6.5 Monte Carlo estimates
243(12)
6.5.1 Basic Monte Carlo
243(4)
6.5.2 Variance reduction methods
247(3)
6.5.3 Multilevel Monte Carlo estimates
250(5)
6.6 Application to option pricing
255(4)
6.7 Summary and further reading
259(1)
Exercises
260(3)
Appendix A Probability reminders 263(8)
A.1 Events and probability
263(3)
A.2 Conditional probability
266(2)
A.3 Expectation
268(1)
A.4 Limit theorems
269(1)
A.5 Further reading
270(1)
Appendix B Programming in R 271(28)
B.1 General advice
271(1)
B.2 R as a Calculator
272(10)
B.2.1 Mathematical operations
273(1)
B.2.2 Variables
273(2)
B.2.3 Data types
275(7)
B.3 Programming principles
282(10)
B.3.1 Don't repeat yourself!
283(3)
B.3.2 Divide and conquer!
286(4)
B.3.3 Test your code!
290(2)
B.4 Random number generation
292(2)
B.5 Summary and further reading
294(1)
Exercises
294(5)
Appendix C Answers to the exercises 299(76)
C.1 Answers for
Chapter 1
299(16)
C.2 Answers for
Chapter 2
315(4)
C.3 Answers for
Chapter 3
319(9)
C.4 Answers for
Chapter 4
328(14)
C.5 Answers for
Chapter 5
342(8)
C.6 Answers for
Chapter 6
350(16)
C.7 Answers for Appendix B
366(9)
References 375(4)
Index 379
Jochen Voss, School of Mathematics, University of Leeds, UK.
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