Intermediate Probability (eBook)

A Computational Approach

Marc S. Paolella (Autor)

eBook Download: PDF

2007
430 Seiten
John Wiley & Sons (Verlag)
978-0-470-03505-4 (ISBN)

Infos

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.

Autor

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

Inhalt

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.

Rezensionen

"I thoroughly enjoyed Intermediate Probability. I was so
thrilled with it that I have shared it with some of my colleagues.
They have called it a 'gold mine' of problems and resources, and
describing it as 'amazing.' ... I highly recommend it."
(Journal of the American Statistical Association, September
2009)



"The reader-friendly style of the text itself would make the
book appropriate for self-study or classroom adoption." (MAA
Reviews, December 2007)

Produkt-Details

• Erscheint lt. Verlag: 27.9.2007
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Statistik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
• Themenwelt: Technik
• Schlagworte: Angewandte Wahrscheinlichkeitsrechnung u. Statistik • Applied Probability & Statistics • Probability & Mathematical Statistics • Statistics • Statistik • Wahrscheinlichkeitsrechnung u. mathematische Statistik
• ISBN-10: 0-470-03505-6 / 0470035056
• ISBN-13: 978-0-470-03505-4 / 9780470035054