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Intermediate Probability: A Computational Approach


Description: 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.

Intermediate Probability is the natural extension of the author's Fundamental Probability. It details several highly important topics, from standard ones such as order statistics, multivariate normal, and convergence concepts, to more advanced ones which are usually not addressed at this mathematical level, or have never previously appeared in textbook form. The author adopts a computational approach throughout, allowing the reader to directly implement the methods, thus greatly enhancing the learning experience and clearly illustrating the applicability, strengths, and weaknesses of the theory.

The book:

- Places great emphasis on the numeric computation of convolutions of random variables, via numeric integration, inversion theorems, fast Fourier transforms, saddlepoint approximations, and simulation.
- Provides introductory material to required mathematical topics such as complex numbers, Laplace and Fourier transforms, matrix algebra, confluent hypergeometric functions, digamma functions, and Bessel functions.
- Presents full derivation and numerous computational methods of the stable Paretian and the singly and doubly non-central distributions.
- A whole chapter is dedicated to mean-variance mixtures, NIG, GIG, generalized hyperbolic and numerous related distributions.
- A whole chapter is dedicated to nesting, generalizing, and asymmetric extensions of popular distributions, as have become popular in empirical finance and other applications.
- Provides all essential programming code in Matlab and R.

The user-friendly style of writing and attention to detail means that self-study is easily possible, making the book ideal for senior undergraduate and graduate students of mathematics, statistics, econometrics, finance, insurance, and computer science, as well as researchers and professional statisticians working in these fields.

About the author:

Marc S Paolella, Swiss Banking Institute, University of Zurich, Switzerland. Associate Professor. Previous career includes:  Director of the Institute of Statistics and Econometrics, Kiel University; Statistical Consultant, Colorado State University; Statistical Programmer at Health Economics Research Inc. Waltham. MA USA. Refereed papers in Journal of the American Statistical Association, Journal of Forecasting, Bernoulli, Statistics and Computing, Applied Economics Quarterly.


Contents: Preface.

I Sums of Random Variables.

1 Generating functions.

1.1 The moment generating function.
1.2 Characteristic functions.
1.3 Use of the fast Fourier transform.
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 Xbar and S2 for i.i.d. normal samples.
3.8 Matrix algebra.
3.9 Problems.

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.
4.4 The central limit theorem.
4.5 Problems.

5 Saddlepoint approximations.

5.1 Univariate.
5.2 Multivariate.
5.3 The hypergeometric functions 1F1 and 2F1.
5.4 Problems.

6 Order statistics.

6.1 Distribution theory for i.i.d. samples.
6.2 Further examples.
6.3 Distribution theory for dependent samples.
6.4 Problems.

III More Flexible and Advanced Random Variables.

7 Generalizing and mixing.

7.1 Basic methods of extension.
7.2 Weighted sums of independent random variables.
7.3 Mixtures.
7.4 Problems.

8 The stable Paretian distribution.

8.1 Symmetric stable.
8.2 Asymmetric stable.
8.3 Moments.
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.
9.4 The generalized inverse Gaussian distribution.
9.5 The generalized hyperbolic distribution.
9.6 Properties of the GHyp distribution family.
9.7 Problems.

10 Noncentral distributions.

10.1 Noncentral chi-square.
10.2 Singly and doubly noncentral F.
10.3 Noncentral beta.
10.4 Singly and doubly noncentral t.
10.5 Saddlepoint uniqueness for the doubly noncentral F.
10.6 Problems.

A Notation and distribution tables.

References.

Index.


Reviews "Very enjoyable and instructive reading." (CHOICE, November 2006) "...a very good book which might, I hope, be standard for introductory courses in probability theory during next years." (Zentralbatt MATH, 11th March 2007)


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