Numerical Methods of Statistics

Front Cover
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
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