Statistics and Computing
2010, XVI, 604 p., Hardcover
ISBN: 978-1-4419-5944-7
This second edition contains new material.
Every advance in computer architecture and software tempts statisticians to tackle numerically harder problems. To do so intelligently requires a good working knowledge of numerical analysis. This book equips students to craft their own software and to understand the advantages and disadvantages of different numerical methods. Issues of numerical stability, accurate approximation, computational complexity, and mathematical modeling share the limelight in a broad yet rigorous overview of those parts of numerical analysis most relevant to statisticians. In this second edition, the material on optimization has been completely rewritten. There is now an entire chapter on the MM algorithm in addition to more comprehensive treatments of constrained optimization, penalty and barrier methods, and model selection via the lasso. There is also new material on the Cholesky decomposition, Gram-Schmidt orthogonalization, the QR decomposition, the singular value decomposition, and reproducing kernel Hilbert spaces. The discussions of the bootstrap, permutation testing, independent Monte Carlo, and hidden Markov chains are updated, and a new chapter on advanced MCMC topics introduces students to Markov random fields, reversible jump MCMC, and convergence analysis in Gibbs sampling. Numerical Analysis for Statisticians can serve as a graduate text for a course surveying computational statistics. With a careful selection of topics and appropriate supplementation, it can be used at the undergraduate level. It contains enough material for a graduate course on optimization theory. Because many chapters are nearly self-contained, professional statisticians will also find the book useful as a reference. Kenneth Lange is the Rosenfeld Professor of Computational Genetics in the Departments of Biomathematics and Human Genetics and the Chair of the Department of Human Genetics, all in the UCLA School of Medicine. His research interests include human genetics, population modeling, biomedical imaging, computational statistics, high-dimensional optimization, and applied stochastic processes. Springer previously published his books Mathematical and Statistical Methods for Genetic Analysis, 2nd ed., Applied Probability, and Optimization. He has written over 200 research papers and produced with his UCLA colleague Eric Sobel the computer program Mendel, widely used in statistical genetics.
Recurrence Relations.- Power Series Expansions.- Continued Fraction Expansions.- Asymptotic Expansions.- Solution of Nonlinear Equations.- Vector and Matrix Norms.- Linear Regression and Matrix Inversion.- Eigenvalues and Eigenvectors.- Singular Value Decomposition.- Splines.- Optimization Theory.- The MM Algorithm.- The EM Algorithm.- Newton's Method and Scoring.- Local and Global Convergence.- Advanced Optimization Topics.- Concrete Hilbert Spaces.- Quadrature Methods.- The Fourier Transform.- The Finite Fourier Transform.- Wavelets.- Generating Random Deviates.- Independent Monte Carlo.- Permutation Tests and the Bootstrap.- Finite-State Markov Chains.- Markov Chain Monte Carlo.- Advanced Topics in MCMC.
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Series: Springer Texts in Statistics
2010, XVIII, 582 p., Hardcover
ISBN: 978-1-4419-6826-5
Due: July 29, 2010
Focuses on thinking skills rather than just what formulae to use
Provides motivations, and intuition, rather than detailed proofs
Begins with very simple and basic techniques, and connects theory and applications in entertaining ways
This book offers a comprehensive guide to large sample techniques in statistics. More importantly, it focuses on thinking skills rather than just what formulae to use; it provides motivations, and intuition, rather than detailed proofs; it begins with very simple techniques, and connects theory and applications in entertaining ways. The first five chapters review some of the basic techniques, such as the fundamental epsilon-delta arguments, Taylor expansion, different types of convergence, and inequalities.
The next five chapters discuss limit theorems in specific situations of observational data. Each of the first 10 chapters contains at least one section of case study. The last five chapters are devoted to special areas of applications. The sections of case studies and chapters of applications fully demonstrate how to use methods developed from large sample theory in various, less-than-textbook situations.
The book is supplemented by a large number of exercises, giving the readers plenty of opportunities to practice what they have learned. The book is mostly self-contained with the appendices providing some backgrounds for matrix algebra and mathematical statistics. The book is intended for a wide audience, ranging from senior undergraduate students to researchers with Ph.D. degrees. A first course in mathematical statistics and a course in calculus are prerequisites.
Jiming Jiang is a Professor of Statistics at the University of California, Davis. He is a Fellow of the American Statistical Association and a Fellow of the Institute of Mathematical Statistics. He is the author of another Springer book, Linear and Generalized Linear Mixed Models and Their Applications (2007). Jiming Jiang is a prominent researcher in the fields of mixed effects models, small area estimation and model selection. Most of his research papers have involved large sample techniques. He is currently an Associate Editor of the Annals of Statistics.
The epsilon-delta arguments.- Modes of convergence.- Big O, small o, and the unspecified c.- Asymptotic expansions.- Inequalities.- Sums of independent random variables.- Empirical processes.- Martingales.- Time and spatial series.- Stochastic processes.- Nonparametric statistics.- Mixed effects models.- Small area estimation.- Jackknife and bootstrap.- Markov chain Monte Carlo.
Series: Springer Series in Statistics
2010, XII, 226 p., Hardcover
ISBN: 978-1-4419-5940-9
An excellent introduction to Bayesian theory and methods, while taking an impartial view of their merits relative to the alternative "classical" or "frequentist" approach A very readable presentation of the basic characteristics of statistical inference from a Bayesian and from a frequentist perspective Offers a resolution of one of the most intense scientific debates in the past 250 years
This monograph contributes to the area of comparative statistical inference. Attention is restricted to the important subfield of statistical estimation. The book is intended for an audience having a solid grounding in probability and statistics at the level of the year-long undergraduate course taken by statistics and mathematics majors. The necessary background on Decision Theory and the frequentist and Bayesian approaches to estimation is presented and carefully discussed in Chapters 1?3. The “threshold problem” -- identifying the boundary between Bayes estimators which tend to outperform standard frequentist estimators and Bayes estimators which don’t -- is formulated in an analytically tractable way in Chapter 4. The formulation includes a specific (decision-theory based) criterion for comparing estimators. The centerpiece of the monograph is Chapter 5 in which, under quite general conditions, an explicit solution to the threshold is obtained for the problem of estimating a scalar parameter under squared error loss. The six chapters that follow address a variety of other contexts in which the threshold problem can be productively treated. Included are treatments of the Bayesian consensus problem, the threshold problem for estimation problems involving of multi-dimensional parameters and/or asymmetric loss, the estimation of nonidentifiable parameters, empirical Bayes methods for combining data from ‘similar’ experiments and linear Bayes methods for combining data from ‘related’ experiments. The final chapter provides an overview of the monograph’s highlights and a discussion of areas and problems in need of further research. F. J. Samaniego is a Distinguished Professor of Statistics at the University of California, Davis. He served as Theory and Methods Editor of the Journal of the American Statistical Association (2003-05), was the 2004 recipient of the Davis Prize for Undergraduate Teaching and Scholarly Achievement, and is an elected Fellow of the ASA, the IMS and the RSS and an elected Member of the ISI.
Point estimation from a decision theoretic viewpoint.- An overview of the frequentist approach to estimation.- An overview of the Bayesian approach to estimation.- The threshold problem.- Comparing Bayesian and frequentist estimators of a scalar parameter.- Conjugacy, self consistency, and Bayesian consensus.- Bayesian vs. frequentist shrinkage in multivariate normal problems.- Comparing Bayesian and frequentist estimators under asymmetric loss.- The treatment of nonidentifiable models.- Improving on standard Bayesian and frequentist estimators.- Combining data from "related" experiments.- Fatherly advice.
Series: Bolyai Society Mathematical Studies, Vol. 21
2010, X, 740 p., Hardcover
ISBN: 978-3-642-14443-1
Due: July 19, 2010
Szemeredi's influence on today's mathematics, especially in combinatorics, additive number theory, and theoretical computer science, is enormous. This volume is a celebration of Szemeredi's achievements and personality, on the occasion of his seventieth birthday. It exemplifies his extraordinary vision and unique way of thinking. A number of colleagues and friends, all top authorities in their fields, have contributed their latest research papers to this volume. The topics include extension and applications of the regularity lemma, the existence of k-term arithmetic progressions in various subsets of the integers, extremal problems in hypergraphs theory, and random graphs, all of them beautiful, Szemeredi type mathematics. It also contains published accounts of the first two, very original and highly successful Polymath projects, one led by Tim Gowers and the other by Terry Tao.
J. Nesetril, P. Ossona De Mendez: Extremal Problems For Sparse Graphs.- Mei-Chu Chang: An Estimate of Incomplete Mixed Character Sums.- Michael A. Nielsen: Introduction to the Polymath Project.- Jozsef Beck: Super-Uniformity of the Typical Billiard Path.- M. T. Keller, N. Streib, W. T. Trotter: Online Linear Discrepancy Of Partially Ordered Sets.- Luca Trevisan: Pseudorandomness in Computer Science and in Additive Combinatorics.- Bela Bollobas and Oliver Riordan: Percolation on self-dual polygon configurations.- L. Lovasz, B. Szegedy: Regularity Partitions and the Topology on Graphons.- Noga Alon: Universality, Tolerance, Chaos and Order.- Ben Green And Terence Tao: An Arithmetic Regularity Lemma, An Associated Counting Lemma, And Applications.- Ben Green And Terence Tao: Yet another proof of Szemeredi's theorem.- D.H.J. POLYMATH: Density Hales-Jewett And Moser Numbers.- Imre Ruzsa: Towards a noncommutative Plunnecke-type inequality.- Tim Gowers: Polymath And The Density Hales-Jewett Theorem.- Joel Spencer: Quasirandom Multitype Graphs.- V. Rodl and A. Rucinski Dirac-type questions for hypergraphs.- Hoi Nguyen and Van Vu: Squares In Sumsets.- Jean Bourgain: On Exponential Sums in Finite Fields.- Y. Kohayakawa, V. Rodl, M. Schacht, J. Skokan: On The Triangle Removal Lemma For Subgraphs Of Subgraphs Of Sparse Pseudorandom Graphs.- Andras Hajnal: My early encounters with Szemeredi.- J. Fox, F. Frati, J. Pach, R. Pinchasi: Crossings between curves with many tangancies .- J. Pintz: Are there arbitrarily long arithmetic progressions in the sequence of twin primes?.- M. Loebl, B. Reed, A. Scott, A. Thomason, S. Thomasse: Almost all H-free graphs have the Erdos-Hajnal property.
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Series: Undergraduate Texts in Mathematics
2010, XII, 328 p. 154 illus., 77 in color., Hardcover
ISBN: 978-1-4419-7287-3
Due: August 29, 2010
The solution of the cubic equation and Newton's method for approximating the zeroes of any polynomial
Expanded treatments of the Schwarz reflection principle and of the mapping properties of analytic functions on closed domains
An introduction to Schwarz?Christoffel transformations and to Dirichlet series
A streamlined proof of the prime number theorem, and more
This unusual and lively textbook offers a clear and intuitive approach to the classical and beautiful theory of complex variables. With very little dependence on advanced concepts from several-variable calculus and topology, the text focuses on the authentic complex-variable ideas and techniques. Notable additions to "Complex Analysis, Third Edition," include: ? The solution of the cubic equation and Newton’s method for approximating the zeroes of any polynomial; ? Expanded treatments of the Schwarz reflection principle and of the mapping properties of analytic functions on closed domains; ? An introduction to Schwarz-Christoffel transformations and to Dirichlet series; ? A streamlined proof of the prime number theorem, and more. Accessible to students at their early stages of mathematical study, this full first year course in complex analysis offers new and interesting motivations for classical results and introduces related topics stressing motivation and technique. Numerous illustrations, examples, and now 300 exercises, enrich the text. Students who master this textbook will emerge with an excellent grounding in complex analysis, and a solid understanding of its wide applicability.
Preface.- The Complex Numbers.- Functions of the Complex Variable z.- Analytic Functions.- Line Integrals and Entire Functions.- Properties of Entire Functions.- Properties of Analytic Functions.- Further Properties of Analytic Functions; 8. Simply Connected Domains.- Isolated Sigularities of an Analytic Function.- The Residue Theorem.- Applications of The Residue Theorem to the Evaluation of Integrals Sums.- Further Contour Integral Techniques.- Introduction to Conformal Mapping.- The Riemann Mapping Theorem.- Maximum-Modulus Theorems for Unbounded Domains.- Harmonic Functions.- Different Forms of Analytic Functions.- Analytic Continuation; The Gamma and Zeta Functions.- Applications to Other Areas of Mathematics.- Appendices.- Answers.- Bibliography.- Index.