Statistical computing / William J. Kennedy, Jr., James E. Gentle.

Format:

Book

Author/Creator:

Kennedy, William J. (William Jo), 1936-

Gentle, James E., 1943- author.

Series:

Statistics, textbooks and monographs ; v. 33.

Statistics, textbooks and monographs ; v. 33

Language:

English

Subjects (All):

Mathematical statistics--Data processing.

Mathematical statistics.

Physical Description:

xi, 591 pages ; 24 cm.

Place of Publication:

New York : M. Dekker, [1980]

Summary:

In this convenient textbook and reference work, the reader will find an introduction to statistical computing and a critical, balanced presentation of the algorithms and computational methods currently in use. Emphasizing the most accurate and widely used of these methods, the book thoroughly describes the algorithms that have been incorporated into the leading software systems of today, and discusses techniques for implementing algorithms in a computer. Statistical Computing contains the detail that researchers need, in the form of a textbook that gives advanced students a broad understanding of the subject, even in its most sophisticated aspects. Complete with exercises and extensive reference lists, Statistical Computing can be applied to a one-semester course for graduate students in statistics, mathematics, computer science, and any field in which numerical methods and algorithms are used in statistical data analyses.

Contents:

1.4 Presentation of Algorithms 5

2 Computer Organization 7

2.2 Components of the Digital Computer System 7

2.3 Representation of Numeric Values 11

2.3.1 Integer Mode Representation 11

2.3.2 Representation in Floating-Point Mode 12

2.4 Floating- and Fixed-Point Arithmetic 14

2.4.1 Floating-Point Arithmetic Operations 14

2.4.2 Fixed-Point Arithmetic Operations 18

3 Error in Floating-Point Computation 23

3.2 Types of Error 24

3.3 Error Due to Approximation Imposed by the Computer 25

3.4 Analyzing Error in a Finite Process 26

3.5 Rounding Error in Floating-Point Computations 27

3.6 Rounding Error in Two Common Floating-Point Calculations 31

3.7 Condition and Numerical Stability 33

3.8 Other Methods of Assessing Error in Computation 37

4 Programming and Statistical Software 45

4.1 Programming Languages: Introduction 46

4.2 Components of Programming Languages 48

4.2.1 Data Types 48

4.2.2 Data Structures 49

4.2.3 Syntax 50

4.2.4 Control Structures 51

4.3 Program Development 51

4.4 Statistical Software 56

5 Approximating Probabilities and Percentage Points in Selected Probability Distributions 61

5.1.1 Probability Distributions 63

5.1.2 Accuracy Considerations 64

5.2 General Methods in Approximation 66

5.2.1 Approximate Transformation of Random Variables 67

5.2.2 Closed Form Approximations 68

5.2.3 General Series Expansion 69

5.2.4 Exact Relationship Between Distributions 72

5.2.5 Numerical Root Finding 72

5.2.6 Continued Fractions 75

5.2.7 Gaussian Quadrature 80

5.2.8 Newton-Cotes Quadrature 86

5.3 The Normal Distribution 89

5.3.1 Normal Probabilities 90

5.3.2 Normal Percentage Points 93

5.4 Student's t Distribution 96

5.4.1 t Probabilities 96

5.4.2 t-Percentage Points 100

5.5 The Beta Distribution 103

5.5.1 Evaluating the Incomplete Beta Function 104

5.5.2 Inverting the Incomplete Beta Function 110

5.6 F Distribution 112

5.6.1 F Probabilities 112

5.6.2 F Percentage Points 116

5.7 Chi-Square Distribution 116

5.7.1 Chi-Square Probabilities 116

5.7.2 Chi-Square Percentage Points 118

6 Random Numbers: Generation, Tests and Applications 133

6.2 Generation of Uniform Random Numbers 136

6.2.1 Congruential Methods 136

6.2.2 Feedback Shift Register Methods 150

6.2.3 Coupled Generators 162

6.2.4 Portable Generators 165

6.3 Tests of Random Number Generators 166

6.3.1 Theoretical Tests 166

6.3.2 Empirical Tests 169

6.3.3 Selecting a Random Number Generator 175

6.4 General Techniques for Generation of Nonuniform Random Deviates 176

6.4.1 Use of the Cumulative Distribution Function 176

6.4.2 Use of Mixtures of Distributions 179

6.4.3 Rejection Methods 184

6.4.4 Table Sampling Methods for Discrete Distributions 192

6.4.5 The Alias Method for Discrete Distributions 197

6.5 Generation of Variates from Specific Distributions 200

6.5.1 The Normal Distribution 201

6.5.2 The Gamma Distribution 209

6.5.3 The Beta Distribution 216

6.5.4 The F, t, and Chi-Square Distributions 219

6.5.5 The Binomial Distribution 221

6.5.6 The Poisson Distribution 223

6.5.7 Distribution of Order Statistics 225

6.5.8 Some Other Univariate Distributions 228

6.5.9 The Multivariate Normal Distribution 228

6.5.10 Some Other Multivariate Distributions 231

6.6 Applications 233

6.6.1 The Monte Carlo Method 233

6.6.2 Sampling and Randomization 236

7 Selected Computational Methods in Linear Algebra 265

7.2 Methods Based on Orthogonal Transformations 266

7.2.1 Householder Transformations 266

7.2.2 Givens Transformations 273

7.2.3 The Modified Gram-Schmidt Method 276

7.2.4 Singular-value Decomposition 278

7.3 Gaussian Elimination and the Sweep Operator 286

7.4 Cholesky Decomposition and Rank-One Update 294

8 Computational Methods for Multiple Linear Regression Analysis 313

8.1.1 Methods Using Orthogonal Triangularization of X 314

8.1.2 Sweep Operations and Normal Equations 321

8.1.3 Checking Programs, Computed Results and Improving Solutions Iteratively 326

8.2 Regression Model Building 331

8.2.1 All Possible Regressions 332

8.2.2 Stepwise Regression 335

8.2.3 Other Methods 340

8.2.4 A Special Case

Polynomial Models 342

8.3 Multiple Regression Under Linear Restrictions 347

8.3.1 Linear Equality Restrictions 348

8.3.2 Linear Inequality Restrictions 352

9 Computational Methods for Classification Models 373

9.1.1 Fixed-effects Models 374

9.1.2 Restrictions on Models and Constraints on Solutions 376

9.1.3 Reductions in Sums of Squares 378

9.2 The Special Case of Balance and Completeness for Fixed-Effects Models 382

9.2.2 Computer-related Considerations in the Special Case 387

9.2.3 Analysis of Covariance 391

9.3 The General Problem for Fixed-Effects Models 395

9.3.1 Estimable Functions 396

9.3.2 Selection Criterion 400

9.4 Computing Expected Mean Squares and Estimates of Variance Components 406

9.4.1 Computing Expected Mean Squares 407

9.4.2 Variance Component Estimation 410

10 Unconstrained Optimization and Nonlinear Regression 425

10.1.1 Iteration 427

10.1.2 Function Minima 427

10.1.3 Step Direction 429

10.1.4 Step Size 431

10.1.5 Convergence of the Iterative Methods 435

10.1.6 Termination of Iteration 436

10.2 Methods for Unconstrained Minimization 437

10.2.1 Method of Steepest Descent 438

10.2.2 Newton's Method and Some Modifications 442

10.2.3 Quasi-Newton Methods 451

10.2.4 Conjugate Gradient Method 460

10.2.5 Conjugate Direction Method 469

10.2.6 Other Derivative-Free Methods 473

10.3 Computational Methods in Nonlinear Regression 475

10.3.1 Newton's Method for the Nonlinear Regression Problem 478

10.3.2 The Modified Gauss-Newton Method 480

10.3.3 The Levenberg-Marquardt Modification of Gauss-Newton 483

10.3.4 Alternative Gradient Methods 485

10.3.5 Minimization Without Derivatives 487

11 Model Fitting Based on Criteria Other Than Least Squares 513

11.2 Minimum L[subscript p] Norm Estimators 514

11.2.1 L[subscript 1] Estimation 515

11.2.2 L[subscript infinity] Estimation 525

11.2.3 Other L[subscript p] Estimators 528

11.3 Other Robust Estimators 532

11.4 Biased Estimation 539

11.5 Robust Nonlinear Regression 544

12 Selected Multivariate Methods 561

12.2 Canonical Correlations 561

12.3 Principal Components 566

12.4 Factor Analysis 567

12.5 Multivariate Analysis of Variance 574.

Notes:

Includes bibliographies and index.

ISBN:

0824768981

OCLC:

6042923