Numerical Analysis for StatisticiansEvery 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. |
Contents
1 Recurrence Relations | 1 |
2 Power Series Expansions | 13 |
3 Continued Fraction Expansions | 26 |
4 Asymptotic Expansions | 39 |
5 Solution of Nonlinear Equations | 55 |
6 Vector and Matrix Norms | 77 |
7 Linear Regression and MatrixInversion | 92 |
8 Eigenvalues and Eigenvectors | 113 |
16 Advanced Optimization Topics | 297 |
17 Concrete Hilbert Spaces | 333 |
18 Quadrature Methods | 363 |
19 The Fourier Transform | 378 |
20 The Finite Fourier Transform | 395 |
21 Wavelets | 412 |
22 Generating Random Deviates | 431 |
23 Independent Monte Carlo | 459 |
9 Singular Value Decomposition | 129 |
10 Splines | 143 |
11 Optimization Theory | 156 |
12 The MM Algorithm | 189 |
13 The EM Algorithm | 223 |
14 Newtons Method and Scoring | 248 |
15 Local and Global Convergence | 277 |
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Common terms and phrases
algorithm applied approximation asymptotic binomial bootstrap Chapter Cholesky decomposition coefficients components compute condition Consider constant constraints continued fraction convergence convex decomposition defined Demonstrate density derivatives deviates diagonal entries differential eigenvalues eigenvectors EM algorithm equation Example expansion exponential finite formula Fourier transform function f(x gamma Gibbs sampling Hilbert space Hint identity implies independent inequality inner product integral interval inverse iterates linear loglikelihood Markov chain matrix matrix norm mean minimizing minimum Monte Carlo multivariate Newton's method nonnegative norm Numerical Analysis observed optimization orthogonal orthogonal matrix orthonormal parameter Pc(x permutation Poisson polynomials positive definite probability Problem Proof Proposition prove quadratic random variable regression resampling satisfies sequence Show Springer Springer Science+Business Media Stat statistics Suppose surrogate function symmetric symmetric matrix theorem tion update variance vector wavelets xn+1 ΣΣ