Intermediate Probability A Computational Approach

by Paolella, Marc S.

• ISBN13: 9780470026373
• ISBN10: 0470026375
• Edition: 1st
• Format: Hardcover
• Copyright: 2007-10-08
• Publisher: Wiley-Interscience

Summary

Many of the traditional and older advanced texts do not cover the newer topics in probability such as Paretian distribution and noncentral distributions and saddlepoint approximation. The material is only covered in research monographs or in journal articles. This text introduces these topics in the context of real-life examples and making full use of the available computer software.The contents include a highly accessible introduction to inversion theorems and their numerical implementation, convolution of random variables, distribution approximations, and the method of saddlepoint approximation (SPA); plus an overview of order statistics (with an introduction to extreme value theory) and the multivariate normal distribution, respectively. Other advanced topics cover the ideas of nesting, generalizing, asymmetric extensions and mixtures; the stable Paretian distribution, with emphasis on its computation, basic properties, and uses; the (generalized) inverse Gaussian and (generalized) hyperbolic distributions, and their connections; and noncentral distributions and quadratic forms.

Author Biography

Marc S Paolella, Professor of Empirical Finance, Swiss Banking Institute, University of Zurich, Switzerland.

Table of Contents

Preface

Sums of Random Variables

Generating functions

The moment generating function

Moments and the m.g.f

The cumulant generating function

Uniqueness of the m.g.f

Vector m.g.f

Characteristic functions

Complex numbers

Laplace transforms

Existence of the Laplace transform

Inverse Laplace transform

Basic properties of characteristic functions

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

If the m.g.f. exists on a neighbourhood of zero

The m.g.f. and c.f. for nonnegative X

Inversion formulae for mass and density functions

Inversion formulae for the c.d.f

Use of the fast Fourier transform

Fourier series

Discrete and fast Fourier transforms

Applying the FFT to c.f. inversion

Multivariate case

Problems

Sums and other functions of several random variables

Weighted sums of independent random variables

Exact integral expressions for functions of two continuous random variables

Approximating the mean and variance

Problems

The multivariate normal distribution

Vector expectation and variance

Basic properties of the multivariate normal

Density and moment generating function

Simulation and c.d.f. calculation

Marginal and conditional normal distributions

Partial correlation

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

Matrix algebra

Problems

II Asymptotics and Other Approximations

Convergence concepts

Inequalities for random variables

Convergence of sequences of sets

Convergence of sequences of random variables

Convergence in probability

Almost sure convergence

Convergence in r-mean

Convergence in distribution

The central limit theorem

Problems

Saddlepoint approximations

Univariate

Density saddlepoint approximation

Saddlepoint approximation to the c.d.f

Detailed illustration: the normal{Laplace sum

Multivariate

Conditional distributions

Bivariate c.d.f. approximation

Marginal distributions

The hypergeometric functions 1F1 and 2F1

Problems

Order statistics

Distribution theory for i.i.d. samples

Univariate

Multivariate

Sample range and midrange

Further examples

Distribution theory for dependent samples

Problems

III More Flexible and Advanced Random Variables

Generalizing and mixing

Basic methods of extension

Nesting and generalizing constants

Extension to the real line

Transformations

Invention of ?exible forms

Weighted sums of independent random variables

Mixtures

Countable mixtures

Continuous mixtures

Problems

The stable Paretian distribution

Symmetric stable

Asymmetric stable

Moments

Mean

Fractional absolute moment proof I

Fractional absolute moment proof II

Simulation

Generalized central limit theorem

Generalized inverse Gaussian and generalized hyperbolic distributions

Introduction

The modi?ed Bessel function of the third kind

Mixtures of normal distributions

Mixture mechanics

Moments and generating functions

The generalized inverse Gaussian distribution

De?nition and general formulae

The subfamilies of the GIG distribution fam

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