An Introduction to Statistical Computing

A Simulation-based Approach

Jochen Voss

• Publisher: Wiley
• Year: 2013
• ISBN:
  9781118728024
• Edition: 1
• Topic: Mathematik
• Subtopic: Wahrscheinlichkeitsrechnung & Statistiken

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 environmentR); 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.

1

Random number generation

The topic of this book is the study of statistical models using computer simulations. Here we use the term ‘statistical models’ to mean any mathematical models which include a random component. Our interest in this chapter and the next is in simulation of the random component of these models. The basic building block of such simulations is the ability to generate random numbers on a computer, and this is the topic of the present chapter. Later, in Chapter 2, we will see how the methods from Chapter 1 can be combined to simulate more complicated models.

Generation of random numbers, or more general random objects, on a computer is complicated by the fact that computer programs are inherently deterministic: while the output of computer program may look random, it is obtained by executing the steps of some algorithm and thus is totally predictable. For example the output of a program computing the decimal digits of the number

(the ratio between the perimeter and diameter of a circle) looks random at first sight, but of course π is not random at all! The output can only start with the string of digits given above and running the program twice will give the same output twice.

We will split the problem of generating random numbers into two distinct subproblems: first we will study the problem of generating any randomness at all, concentrating on the simple case of generating independent random numbers, uniformly distributed on the interval [0, 1]. This problem and related concerns will be discussed in Section 1.1. In the following sections, starting with Section 1.2, we will study the generation of random numbers from different distributions, using the independent, uniformly distributed random numbers obtained in the previous step as a basis.

1.1 Pseudo random number generators

There are ...