Numerical Analysis for Statisticians

Kenneth Lange0-387-94979-8

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 is intended to equip 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 relevant to statisticians. Although the bulk of the book covers traditional topics from linear algebra, optimization theory, numerical integration, and Fourier analysis, several chapters highlight recent statistical developments such as wavelets, the bootstrap, hidden Markov chains, and Markov chain Monte Carlo methods. These computationally intensive methods are revolutionizing statistics. Numerical Analysis for Statisticians can serve as a graduate text for either a one or a two-semester course surveying computational statistics. With a careful selection of topics and appropriate supplementation, it can even be used at the undergraduate level. It contains enough material on optimization theory alone for a one-semester graduate course. Students mastering a substantial part of the text will be well prepared for the numerical parts of advanced topics courses in statistics. Because many of the chapters nearly self-contained, professional statisticians will also find the book useful as a reference. Kenneth Lange is Professor of Biomathematics and Human Genetics at the UCLA School of Medicine. At various times during his career, he has held appointments at the University of New Hampshire, MIT, Harvard, and the University of Michigan. While at the University of Michigan, he was the Parmacia Upjohn Foundation, Professor of Biostatistics. His research interests include human genetics, population modeling, biomedical imaging, computational statistics, and applied stochastic processes. Also available by Kenneth Lange: Mathematical and Statistical Methods for Genetic Analysis, Springer-Verlag New York Inc., 1997, 265 pp., Cloth, ISBN 0-387-949097.

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Table of contents :
Cover Page……Page 0
Title Page……Page 4
Edition Info……Page 5
2 Power Series Expansions……Page 10
5 Solution of Nonlinear Equations……Page 11
9 Splines……Page 12
14 Constrained Optimization……Page 13
18 The Finite Fourier Transform……Page 14
23 Finite-State Markov Chains……Page 15
Index……Page 16
Preface……Page 6
1 Recurrence Relations……Page 17
2 Power Series Expansions……Page 28
3 Continued Fraction Expansions……Page 41
4 Asymptotic Expansions……Page 53
5 Solution of Nonlinear Equations……Page 69
6 Vector and Matrix Norms……Page 84
7 Linear Regression and Matrix Inversion……Page 95
8 Eigenvalues and Eigenvectors……Page 108
9 Splines……Page 119
10 The EM Algorithm……Page 131
11 Newton’s Method and Scoring……Page 146
12 Variations on the EM Theme……Page 159
13 Convergence of Optimization Algorithms……Page 176
14 Constrained Optimization……Page 193
15 Concrete Hilbert Spaces……Page 207
16 Quadrature Methods……Page 223
17 The Fourier Transform……Page 237
18 The Finite Fourier Transform……Page 251
19 Wavelets……Page 268
20 Generating Random Deviates……Page 285
21 Independent Monte Carlo……Page 302
22 Bootstrap Calculations……Page 315
23 Finite-State Markov Chains……Page 330
24 Markov Chain Monte Carlo……Page 346
A,B……Page 360
C……Page 361
E……Page 362
F……Page 363
G,H……Page 364
K,L……Page 365
M……Page 366
N,O……Page 367
R……Page 368
T……Page 370
U,V,W……Page 371