# Numerical Methods of Statistics

Cambridge University Press, Feb 5, 2001 - Computers - 428 pages
This book explains how computer software is designed to perform the tasks required for sophisticated statistical analysis. For statisticians, it examines the nitty-gritty computational problems behind statistical methods. For mathematicians and computer scientists, it looks at the application of mathematical tools to statistical problems. The first half of the book offers a basic background in numerical analysis that emphasizes issues important to statisticians. The next several chapters cover a broad array of statistical tools, such as maximum likelihood and nonlinear regression. The author also treats the application of numerical tools; numerical integration and random number generation are explained in a unified manner reflecting complementary views of Monte Carlo methods. The book concludes with an examination of sorting, FFT and the application of other "fast" algorithms to statistics. Each chapter contains exercises that range in difficulty as well as examples of the methods at work. Most of the examples are accompanied by demonstration code available from the author's home page.

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### Contents

 Algorithms and Computers 1 12 Computers 3 13 Software and Computer Languages 5 14 Data Structures 8 15 Programming Practice 9 References 10 Computer Arithmetic 11 22 Positional Number Systems 12
 Numerical Differentiation 184 87 Minimization and Nonlinear Equations 187 88 Condition and Scaling 192 89 Implementation 193 Programs and Demonstrations 194 Exercises 196 References 198 Maximum Likelihood and Nonlinear Regression 199

 23 Fixed Point Arithmetic 15 24 Floating Point Representations 18 25 Living with Floating Point Inaccuracies 21 26 The Pale and Beyond 26 27 Conditioned Problems and Stable Algorithms 30 Programs and Demonstrations 32 Exercises 33 References 35 Matrices and Linear Equations 37 32 Matrix Operations 38 33 Solving Triangular Systems 40 34 Gaussian Elimination 41 35 Cholesky Decomposition 47 36 Matrix Norms 50 37 Accuracy and Conditioning 52 Programs and Demonstrations 56 Exercises 58 References 59 More Methods for Solving Linear Equations 61 43 Banded Matrices 65 44 Applications to ARMA TimeSeries Models 67 45 Toeplitz Systems 70 46 Sparse Matrices 73 47 Iterative Methods 76 Programs and Demonstrations 78 References 80 Regression Computations 82 52 Condition of the Regression Problem 84 53 Solving the Normal Equations 87 54 GramSchmidt Orthogonalization 88 55 Householder Transformations 91 56 Householder Transformations for Least Squares 92 57 Givens Transformations 95 59 Regression Diagnostics 98 510 Hypothesis Tests 100 511 Conjugate Gradient Methods 103 512 Doolittle the Sweep and AH Possible Regressions 106 513 Comments 108 Exercises 109 References 111 Eigenproblems 114 63 Power Methods 116 64 The Symmetric Eigenproblem and Tridiagonalization 119 65 The QR Algorithm 121 66 Singular Value Decomposition 123 67 Applications 126 68 Complex Singular Value Decomposition 130 Programs and Demonstrations 132 Exercises 133 References 136 Functions Interpolation Smoothing and Approximation 137 72 Interpolation 139 73 Interpolating Splines 142 Smoothing and Regression 145 75 Mathematical Approximation 148 76 Practical Approximation Techniques 152 77 Computing Probability Functions 155 Programs and Demonstrations 162 Exercises 164 References 168 Introduction to Optimization and Nonlinear Equations 170 Lattice Search Golden Section and Bisection 172 83 Root Finding 175 Stopping and Condition 181 85 Multivariate Newtons Methods 183
 92 Notation and Asymptotic Theory of Maximum Likelihood 200 93 Information Scoring and Variance Estimates 206 94 An Extended Example 208 95 Concentration Iteration and the EM Algorithm 210 96 Multiple Regression in the Context of Maximum Likelihood 216 97 Generalized Linear Models 217 98 Nonlinear Regression 221 99 Parameterizations and Constraints 225 Programs and Demonstrations 229 Exercises 231 References 233 Numerical Integration and Monte Carlo Methods 235 102 Motivating Problems 236 103 OneDimensional Quadrature 242 104 Numerical Integration in Two or More Variables 249 105 Uniform Pseudorandom Variables 256 106 QuasiMonte Carlo Integration 263 107 Strategy and Tactics 268 Programs and Demonstrations 272 Exercises 274 References 276 Generating Random Variables from Other Distributions 279 112 General Methods for Continuous Distributions 280 113 Algorithms for Continuous Distributions 284 114 General Methods for Discrete Distributions 297 115 Algorithms for Discrete Distributions 301 116 Other Randomizations 306 117 Accuracy in Random Number Generation 310 Programs and Demonstrations 313 Exercises 314 References 317 Statistical Methods for Integration and Monte Carlo 319 123 Distributional Tests 326 124 Importance Sampling and Weighted Observations 329 125 Testing Importance Sampling Weights 335 126 Laplace Approximations 337 127 Randomized Quadrature 339 128 SphericalRadial Methods 341 Programs and Demonstrations 346 Exercises 348 References 349 Markov Chain Monte Carlo Methods 351 132 Markov Chains 353 133 Gibbs Sampling 354 134 MetropolisHastings Algorithm 359 135 TimeSeries Analysis 362 136 Adaptive AcceptanceRejection 366 137 Diagnostics 370 Programs and Demonstrations 374 References 376 Sorting and Fast Algorithms 379 143 Sorting Algorithms 381 144 Fast Order Statistics and Related Problems 384 145 Fast Fourier Transform 385 146 Convolutions and the Chirpz Transform 389 147 Statistical Applications of the FFT 391 148 Combinatorial Problems 401 Programs and Demonstrations 405 Exercises 409 References 412 Table of Programs and Demonstrations 415 Author Index 419 Subject Index 425 Copyright

### References to this book

 Bayesian Field TheoryJörg C. LemmLimited preview - 2003
 COMPSTAT 2004 - Proceedings in Computational Statistics: 16th Symposium Held ...Jaromir AntochNo preview available - 2004
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