Intermediate Probability
A Computational Approach
1st Edition
Published on 27. September 2007
430 pages
978-0-470-03505-4 (ISBN)
Description
Intermediate Probability is the natural extension of the author'sFundamental Probability. It details several highly importanttopics, from standard ones such as order statistics, multivariatenormal, and convergence concepts, to more advanced ones which areusually not addressed at this mathematical level, or have neverpreviously appeared in textbook form. The author adopts acomputational approach throughout, allowing the reader to directlyimplement the methods, thus greatly enhancing the learningexperience and clearly illustrating the applicability, strengths,and weaknesses of the theory.
The book:
* Places great emphasis on the numeric computation ofconvolutions of random variables, via numeric integration,inversion theorems, fast Fourier transforms, saddlepointapproximations, and simulation.
* Provides introductory material to required mathematical topicssuch as complex numbers, Laplace and Fourier transforms, matrixalgebra, confluent hypergeometric functions, digamma functions, andBessel functions.
* Presents full derivation and numerous computational methods ofthe stable Paretian and the singly and doubly non-centraldistributions.
* A whole chapter is dedicated to mean-variance mixtures, NIG,GIG, generalized hyperbolic and numerous relateddistributions.
* A whole chapter is dedicated to nesting, generalizing, andasymmetric extensions of popular distributions, as have becomepopular 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 meansthat self-study is easily possible, making the book ideal forsenior undergraduate and graduate students of mathematics,statistics, econometrics, finance, insurance, and computer science,as well as researchers and professional statisticians working inthese fields.
Reviews / Votes
"I thoroughly enjoyed Intermediate Probability. I was sothrilled with it that I have shared it with some of my colleagues.They have called it a 'gold mine' of problems and resources, anddescribing it as 'amazing.' ... I highly recommend it."(Journal of the American Statistical Association, September2009) "The reader-friendly style of the text itself would make thebook appropriate for self-study or classroom adoption." (MAAReviews, December 2007)More details
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Marc S. Paolella
Intermediate Probability - A Computational Approach
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09/2007
Wiley
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08/2007
1st Edition
Wiley
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Person
Marc S Paolella, Professor of Empirical Finance, Swiss Banking Institute, University of Zurich, Switzerland.
Content
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 randomvariables.
2.1 Weighted sums of independent random variables.
2.2 Exact integral expressions for functions of two continuousrandom
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. normalsamples.
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 hyperbolicdistributions.
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.
