Sigma Series in Stochastics -- Volume 3
xii + 170 pages, soft cover, ISBN 978-3-88538-303-1,
This book is a monograph that builds on the path-breaking work of Arthur Dempster and Glenn Shafer, and before them R. A. Fisher, in the field of statistical inference. The main thrust of the book lies around the idea of statistical information. The inferential mechanism that is used in this book to derive information about a parameter in a statistical experiment is new, which is what sets it apart from other books on the topic. This inference principle, which we call assumption-based reasoning, is based on a sound combination of logic and classical probability theory. Some traces of it can already be found in the original writings of Jacob Bernoulli, but this book presents for the first time a much more complete and elaborate descripton of it. In particular, it is shown that assumption-based inference on functional models is a generalization of both Bayesian inference and Fisher's fiducial inference. This is an interesting result regarding the old controversy between these two theories. Our approach provides a new and clear meaning to post-data probabilistic statements about an unknown parameter, for example statements based on the likelihood function. In particular, it is shown that this function cannot, in general, be considered to carry the entire statistical information contained in the experiment.
Information about statistical experiments is described in this monograph by functional models. They indicate how observations are generated in functions of an unknown parameter and stochastic disturbances. In the first part of the book we examine discrete functional models. These models are used to present the basic ideas of assumption-based reasoning in a form that is unhampered by technical difficulties. It is shown how several pieces of information can be combined and how the result can be focused on a question of interest. These operations provide an algebraic flavor to the analysis of statistical information, which is a perspective that is presented here for the first time. Some new preliminary results regarding a decision rule for hypothesis selection are also presented. In the second part of the book we treat several types of continuous models from the standpoint of assumption-based reasoning. This allows us to review and clarify several concepts and difficulties of Fisher's fiducial theory, for example the relation between traditional confidence intervals and fiducial intervals. Our approach also permits to determine the exact role and nature of improper priors in Bayesian inference. Finally, the third part of the book is dedicated to the analysis of linear models with Gaussian perturbations using assumption-based reasoning.
Contents
ISBN: 978-0-470-51886-1
Paperback
344 pages
June 2008
Written by one of the world's leading researchers and writers in the field, Econometric Analysis of Panel Data has become established as the leading textbook for postgraduate courses in panel data. This new edition reflects the rapid developments in the field covering the vast research that has been conducted on panel data since its initial publication. Featuring the most recent empirical examples from panel data literature, data sets are also provided as well as the programs to implement the estimation and testing procedures described in the book. These programs will be made available via an accompanying website which will also contain solutions to end of chapter exercises that will appear in the book.
The text has been fully updated with new material on dynamic panel data models and recent results on non-linear panel models and in particular work on limited dependent variables panel data models.
Contents
Wiley Series in Probability and Statistics)
ISBN: 978-0-470-02842-1
Hardcover
320 pages
June 2008
Multivariable regression models are of fundamental importance in all areas of science in which empirical data must be analyzed. This book proposes a systematic approach to building such models based on standard principles of statistical modeling. The main emphasis is on the fractional polynomial method for modeling the influence of continuous variables in a multivariable context, a topic for which there is no standard approach. Existing options range from very simple step functions to highly complex adaptive methods such as multivariate splines with many knots and penalisation. This new approach, developed in part by the authors over the last decade, is a compromise which promotes interpretable, comprehensible and transportable models.
(hardback)
ISBN-13: 978-0-19-921970-4
Estimated publication date: May 2008
224 pages, 234x156 mm
Series: Oxford Graduate Texts in Mathematics number 18
Description
An up-to-date reference on a topical subject
The author is a well-regarded researcher in the field
Each chapter on filtering contains a historical note
Methods introduced can be applied to more general stochastic partial differential equations
Stochastic Filtering Theory uses probability tools to estimate unobservable stochastic processes that arise in many applied fields including communication, target-tracking, and mathematical finance.
As a topic, Stochastic Filtering Theory has progressed rapidly in recent years. For example, the (branching) particle system representation of the optimal filter has been extensively studied to seek more effective numerical approximations of the optimal filter; the stability of the filter with "incorrect" initial state, as well as the long-term behavior of the optimal filter, has attracted the attention of many researchers; and although still in its infancy, the study of singular filtering models has yielded exciting results.
In this text, Jie Xiong introduces the reader to the basics of Stochastic Filtering Theory before covering these key recent advances. The text is written in a style suitable for graduates in mathematics and engineering with a background in basic probability.
Readership: Graduates and researchers in applied mathematics, engineering and mathematical finance.
Contents
Preface
1. Introduction
2. Brownian motion and martingales
3. Stochastic intervals and Ito's formula
4. Stochastic differential equations
5. Filtering model and Kallianpur-Striebel formula
6. Uniqueness of the solution for Zakai's equation
7. Uniqueness of the solution for the filtering equation
8. Numerical methods
9. Linear filtering
10. Stability of nonlinear filtering
11. Singular filtering
Bibliography
Index
(Hardback)
ISBN-13: 978-0-19-923072-3
Estimated publication date: July 2008
600 pages, 234x156 mm
Series: Oxford Mathematical Monographs
Description
Authored by a leading researcher in the field
A complete basis for General Relativity
Complete proofs or precise references
Highlights open questions
General Relativity has passed all experimental and observational tests to model the motion of isolated bodies with strong gravitational fields, though the mathematical and numerical study of these motions is still in its infancy. It is believed that General Relativity models our cosmos, with a manifold of dimensions possibly greater than four and debatable topology opening a vast field of investigation for mathematicians and physicists alike. Remarkable conjectures have been proposed, many results have been obtained but many fundamental questions remain open. In this monograph, aimed at researchers in mathematics and physics, the author overviews the basic ideas in General Relativity, introduces the necessary mathematics and discusses some of the key open questions in the field.
Readership: Andvanced undergraduates, graduates, and researchers in Applied Mathematics, Physics, Cosmology and Astrophysics.
Contents
Foreword
Acknowledgements
1. Lorentzian Geometry
2. Special Relativity
3. General Relativity and the Einstein Equations
4. Schwarzschild Space-time and Black Holes
5. Cosmology
6. Local Cauchy Problem
7. Constraints
8. Other Hyperbolic-Elliptic systems
9. Relativistic Fluids
10. Kinetic Theory
11. Progressive Waves
12. Global Hyperbolicity and Causality
13. Singularities
14. Stationary Space-times and Black Holes
15. Global Existence Theorems, Asymptotically Euclidean Data
16. Global existence theorems, cosmological case
Appendices
I. Sobolev Spaces
II. Elliptic Systems
III. Second Order Quasidiagonal Systems
IV. General Hyperbolic Systems
V. Cauchy Kovalevski and Fuchs theorems
VI. Conformal Methods
VII. Kaluza Klein Formulas
(Hardback)ISBN-13: 978-0-19-921985-8
(paper)ISBN-13: 978-0-19-921986-5
Estimated publication date: June 2008
480 pages, 234x156 mm
Description
Much-needed update of a classic text
Extensive end-of-chapter notes
Suggestions for further reading for the more avid reader
New chapter on one of the most important developments in number theory and its role in the proof of Fermat's Last Theorem
New to this edition
Revised end-of-chapter notes
New chapter on elliptic curves
An Introduction to the Theory of Numbers by G.H. Hardy and E. M. Wright is found on the reading list of virtually all elementary number theory courses and is widely regarded as the primary and classic text in elementary number theory. Developed under the guidance of D.R. Heath-Brown this Sixth Edition of An Introduction to the Theory of Numbers has been extensively revised and updated to guide today's students through the key milestones and developments in number theory.
Updates include a chapter by J.H. Silverman on one of the most important developments in number theory -- modular elliptic curves and their role in the proof of Fermat's Last Theorem -- a foreword by A. Wiles, and comprehensively updated end-of-chapter notes detailing the key developments in number theory. Suggestions for further reading are also included for the more avid reader
The text retains the style and clarity of previous editions making it highly suitable for undergraduates in mathematics from the first year upwards as well as an essential reference for all number theorists.
Readership: Undergraduates in mathematics, sepcifically number theory and algebra.
Contents
Preface to the sixth edition
Andrew Wiles
Preface to the fifth edition
1. The Series of Primes (1)
2. The Series of Primes (2)
3. Farey Series and a Theorem of Minkowski
4. Irrational Numbers
5. Congruences and Residues
6. Fermat's Theorem and its Consequences
7. General Properties of Congruences
8. Congruences to Composite Moduli
9. The Representation of Numbers by Decimals
10. Continued Fractions
11. Approximation of Irrationals by Rationals
12. The Fundamental Theorem of Arithmetic in k(l), k(i), and k(p)
13. Some Diophantine Equations
14. Quadratic Fields (1)
15. Quadratic Fields (2)
16. The Arithmetical Functions o(n), ƒÊ(n), delta(n), ƒÐ(n), r(n)
17. Generating Functions of Arithmetical Functions
18. The Order of Magnitude of Arithmetical Functions
19. Partitions
20. The Representation of a Number by Two or Four Squares
21. Representation by Cubes and Higher Powers
22. The Series of Primes (3)
23. Kronecker's Theorem
24. Geometry of Numbers
25. Elliptic Curves , Joseph H. Silverman
Appendix
List of Books
Index of Special Symbols and Words
Index of Names
General Index