Intermediate Probability: A Computational Approach.pdf

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Intermediate Probability

Chapter Listing

Contents

Preface

Part I Sums of Random Variables

1 Generating functions

1.1 The moment generating function

1.1.1 Moments and the m.g.f.

1.1.2 The cumulant generating function

1.1.3 Uniqueness of the m.g.f.

1.1.4 Vector m.g.f.

1.2 Characteristic functions

1.2.1 Complex numbers

1.2.2 Laplace transforms

1.2.3 Basic properties of characteristic functions

1.2.4 Relation between the m.g.f. and c.f.

1.2.5 Inversion formulae for mass and density functions

1.2.6 Inversion formulae for the c.d.f.

1.3 Use of the fast Fourier transform

1.3.1 Fourier series

1.3.2 Discrete and fast Fourier transforms

1.3.3 Applying the FFT to c.f. inversion

1.4 Multivariate case

1.5 Problems

2 Sums and other functions of several random variables

2.1 Weighted sums of independent random variables

2.2 Exact integral expressions for functions of two continuous random variables

2.3 Approximating the mean and variance

2.4 Problems

3 The multivariate normal distribution

3.1 Vector expectation and variance

3.2 Basic properties of the multivariate normal

3.3 Density and moment generating function

3.4 Simulation and c.d.f. calculation

3.5 Marginal and conditional normal distributions

3.6 Partial correlation

3.7 Joint distribution of X and S2 for i.i.d. normal samples

3.8 Matrix algebra

3.9 Problems

Part II Asymptotics and Other Approximations

4 Convergence concepts

4.1 Inequalities for random variables

4.2 Convergence of sequences of sets

4.3 Convergence of sequences of random variables

4.3.1 Convergence in probability

4.3.2 Almost sure convergence

4.3.3 Convergence in r-mean

4.3.4 Convergence in distribution

4.4 The central limit theorem

4.5 Problems

5 Saddlepoint approximations

5.1 Univariate

5.1.1 Density saddlepoint approximation

5.1.2 Saddlepoint approximation to the c.d.f.

5.1.3 Detailed illustration: the normal–Laplace sum

5.2 Multivariate

5.2.1 Conditional distributions

5.2.2 Bivariate c.d.f. approximation

5.2.3 Marginal distributions

5.3 The hypergeometric functions 1F1 and 2F1

5.4 Problems

6 Order statistics

6.1 Distribution theory for i.i.d. samples

6.1.1 Univariate

6.1.2 Multivariate

6.1.3 Sample range and midrange

6.2 Further examples

6.3 Distribution theory for dependent samples

6.4 Problems

Part III More Flexible and Advanced Random Variables

7 Generalizing and mixing

7.1 Basic methods of extension

7.1.1 Nesting and generalizing constants

7.1.2 Asymmetric extensions

7.1.3 Extension to the real line

7.1.4 Transformations

7.1.5 Invention of flexible forms

7.2 Weighted sums of independent random variables

7.3 Mixtures

7.3.1 Countable mixtures

7.3.2 Continuous mixtures

7.4 Problems

8 The stable Paretian distribution

8.1 Symmetric stable

8.2 Asymmetric stable

8.3 Moments

8.3.1 Mean

8.3.2 Fractional absolute moment proof I

8.3.3 Fractional absolute moment proof II

8.4 Simulation

8.5 Generalized central limit theorem

9 Generalized inverse Gaussian and generalized hyperbolic distributions

9.1 Introduction

9.2 The modified Bessel function of the third kind

9.3 Mixtures of normal distributions

9.3.1 Mixture mechanics

9.3.2 Moments and generating functions

9.4 The generalized inverse Gaussian distribution

9.4.1 Definition and general formulae

9.4.2 The subfamilies of the GIG distribution family

9.5 The generalized hyperbolic distribution

9.5.1 Definition, parameters and general formulae

9.5.2 The subfamilies of the GHyp distribution family

9.5.3 Limiting cases of GHyp

9.6 Properties of the GHyp distribution family

9.6.1 Location–scale behaviour of GHyp

9.6.2 The parameters of GHyp

9.6.3 Alternative parameterizations of GHyp

9.6.4 The shape triangle

9.6.5 Convolution and infinite divisibility

9.7 Problems

10 Noncentral distributions

10.1 Noncentral chi-square

10.1.1 Derivation

10.1.2 Moments

10.1.3 Computation

10.1.4 Weighted sums of independent central 2 random variables

10.1.5 Weighted sums of independent χ[sup(2)] η[sub(i)],θ[sub(i)]) random variables

10.2 Singly and doubly noncentral F

10.2.1 Derivation

10.2.2 Moments

10.2.3 Exact computation

10.2.4 Approximate computation methods

10.3 Noncentral beta

10.4 Singly and doubly noncentral t

10.4.1 Derivation

10.4.2 Saddlepoint approximation

10.4.3 Moments

10.5 Saddlepoint uniqueness for the doubly noncentral F

10.6 Problems

A Notation and distribution tables

References

Index

Intermediate Probability

Intermediate Probability A Computational Approach Marc S. Paolella Swiss Banking Institute, University of Zurich, Switzerland

Copyright  2007 John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England Telephone (+44) 1243 779777 Email (for orders and customer service enquiries): cs-books@wiley.co.uk Visit our Home Page on www.wileyeurope.com or www.wiley.com All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except under the terms of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London W1T 4LP, UK, without the permission in writing of the Publisher. Requests to the Publisher should be addressed to the Permissions Department, John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England, or emailed to permreq@wiley.co.uk, or faxed to (+44) 1243 770620. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold on the understanding that the Publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional should be sought. Other Wiley Editorial Ofﬁces John Wiley & Sons Inc., 111 River Street, Hoboken, NJ 07030, USA Jossey-Bass, 989 Market Street, San Francisco, CA 94103-1741, USA Wiley-VCH Verlag GmbH, Boschstr. 12, D-69469 Weinheim, Germany John Wiley & Sons Australia Ltd, 42 McDougall Street, Milton, Queensland 4064, Australia John Wiley & Sons (Asia) Pte Ltd, 2 Clementi Loop #02-01, Jin Xing Distripark, Singapore 129809 John Wiley & Sons Canada Ltd, 6045 Freemont Blvd, Mississauga, Ontario, L5R 4J3, Canada Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. Anniversary Logo Design: Richard J. Paciﬁco Library of Congress Cataloging-in-Publication Data Paolella, Marc S. Intermediate probability : a computational approach / Marc S. Paolella. p. cm. ISBN 978-0-470-02637-3 (cloth) 1. Distribution (Probability theory)–Mathematical models. 2. Probabilities. I. Title. QA273.6.P36 2007 519.2 – dc22 2007020127 British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN-13: 978-0-470-02637-3 Typeset in 10/12 Times by Laserwords Private Limited, Chennai, India Printed and bound in Great Britain by Antony Rowe Ltd, Chippenham, Wiltshire This book is printed on acid-free paper responsibly manufactured from sustainable forestry in which at least two trees are planted for each one used for paper production.

Chapter Listing Preface Part I Sums of Random Variables 1 Generating functions 2 Sums and other functions of several random variables 3 The multivariate normal distribution Part II Asymptotics and Other Approximations 4 Convergence concepts 5 Saddlepoint approximations 6 Order statistics Part III More Flexible and Advanced Random Variables 7 Generalizing and mixing 8 The stable Paretian distribution 9 Generalized inverse Gaussian and generalized hyperbolic distributions 10 Noncentral distributions Appendix A Notation and distribution tables References Index

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