Statistical Computing

Preface
1/ Introduction
Orientation
Purpose
Prerequisites
Presentation of Algorithms
2/ Computer Organization
2.1. Introduction
2.2. Components of the Digital Computer System
2.3 Representation of Numeric Values
2.3.1 Integer Mode Representation
2.3.2 Representation in Floating-Point Mode
2.4 Floating- and Fixed- Point Arithmetic
2.4.1 Floating-Point Arithmetic Operations
2.4.2 Fixed-Point Arithmetic Operations
Exercises
References
3/ Error In Floating-Point Computation
3.1 Introduction
3.2 Types of Error
3.3 Error Due to Approximation Imposed by the Computer
3.4 Analyzing Error in a Finite Process
3.5 Rounding Error in Floating-Point Computations
3.6 Rounding Error in Two Common Floating-Point Calculations
3.7 Condition and Numerical Stability
3.8 Other Methods of Assessing Error in computation
3.9 Summary
Exercises
References
4/ Programming and Statistical Software
4.1 Programming Languages: Introduction
4.2 Components of Programming Languages
4.2.1 Data Types
4.2.2 Data Structures
4.2.3 Syntax
4.2.4 Control Structures
4.3 Program Development
4.4 Statistical Software
References and Further Readings
5/ Approximating Probabilities and Percentage Selected Probability Distributions
5.1 Notation and General Considerations
5.1.1 Probability Distributions
5.1.2 Accuracy Considerations
5.2 General Methods in Approximation
5.2.1 Approximate Transformation of Random Variables
5.2.2 Closed Form Approximations
5.2.3 General Series Expansion
5.2.4 Exact Relationship Between Distributions
5.2.5 Numerical Root Finding
5.2.6 Continued Fractions
5.2.7 Gaussian Quadrature
5.2.8 Newton-Cotes Quadrature
5.3 The Normal Distribution
5.3.1 Normal Probabilities
5.3.2 Normal Percentage Points
5.4 Student’s t Distribution
5.4.1 t Probabilities
5.4.2 t-Percentage Points
5.5 The Beta Distributions
5.5.1 Evaluating the Incomplete Beta Function
5.5.2 Inverting the Incomplete Beta Function
5.6 F Distribution
5.6.1 F Probabilities
5.6.2 F Percentage Points
5.7 Chi-Square Distribution
5.7.1 Chi-Square Probabilities
5.7.2 Chi-Square Percentage Points
Exercises
References and Further Readings
6/ Random Numbers: Generation, Tests and Applications
6.1 Introduction
6.2 Generation of Uniform Random Numbers
6.2.1 Congruential Methods
6.2.2 Feedback Shift Register Methods
6.2.3 Coupled Generators
6.2.4 Portable Generators
6.3 Tests of Random Number Generators
6.3.1 Theoretical Tests
6.3.2 Empirical Tests
6.3.3 Selecting a Random Number Generator
6.4 General Techniques for Generation of Nonuniform Random Deviates
6.4.1 Use of the Cumulative Distribution Function
6.4.2 Use of Mixtures of Distributions
6.4.3 Rejection Methods
6.4.4 Table Sampling Methods for Discrete Distributions
6.4.5 The Alias Method for Discrete Distributions
6.5 generation of Variates from Specific Distributions
6.5.1 The Normal Distribution
6.5.2 The Gamma Distribution
6.5.3 The Beta Distribution
6.5.4 The F, t, and Chi-Square Distributions
6.5.5 The Binomial Distribution
6.5.6 The Poisson Distribution
6.5.7 Distribution of Order Statistics
6.5.8 Some Other Univariate Distributions
6.5.9 The Multivariate Normal Distribution
6.5.10 Some Other Multivariate Distributions
6.6 Applications
6.6.1 The Monte Carlo Method
6.6.2 Sampling and Randomization
Exercises
References and Further Readings
7/ Selected Computational Methods In Linear Algebra
7.1 Introduction
7.2 Methods Bases on Orthogonal Transformations
7.2.1 Householder Transformations
7.2.2 Givens Transformations
7.2.3 The Modified Gram-Schmidt Method
7.2.4 Singular-value Decomposition
7.3 Gaussian Elimination and the Sweep Operator
7.4 Cholesky Decomposition and Rank-One Update
Exercises
References and Further Readings
8/ Computational Methods for Multiple Linear Regression Analysis
8.1 Basic Computational Methods
8.1.1 Methods Using Orthogonal Triangularization of X
8.1.2 Sweep Operations and Normal Equations
8.1.3 Checking Programs, Computed Results and Improving Solutions Iteratively
8.2 Regression Model Building
8.2.1 All Possible Regressions
8.2.2 Stepwise Regression
8.2.3 Other Methods
8.2.4 A Special Case—Polynomial Models
8.3 Multiple Regression Under Linear Restrictions
8.3.1 Linear Equality Restrictions
8.3.2 Linear Inequality Restrictions
Exercises
References and Further Readings
9/ Computational Methods For Classification Models
9.1 Introduction
9.1.1 Fixed-effects Models
9.1.2 Restrictions on Models and Constraints on Solutions
9.1.3 Reductions in Sums of squares
9.1.4 An Example
9.2 The Special Case of Balance and Completeness for Fixed-Effects Models
9.2.1 Basic definitions and considerations
9.2.2 Computer-related Considerations in the Special Case
9.2.3 Analysis of Covariance
9.3 The General Problem for Fixed-Effects Models
9.3.1 Estimable Functions
9.3.2 Selection Criterion 1
9.3.3 Selection Criterion 2
9.3.4 Summary
9.4 Computing Expected Mean Squares and Estimates of Variance Components
9.4.1 Computing Expected Mean Squares
9.4.2 Variance Component Estimation
Exercises
References and Further Readings
10/ Unconstrained Optimization and Nonlinear Regression
10.1 Preliminaries
10.1.1 Iteration
10.1.2 Function Minima
10.1.3 Step Direction
10.1.4 Step Size
10.1.5 Convergence of the Iterative Methods
10.1.6 Termination of Iteration
10.2 Methods for Unconstrained Minimization
10.2.1 Method of Steepest Descent
10.2.2 Newton’s Method and Some Modifications
10.2.3 Quasi-Newton Methods
10.2.4 Conjugate Gradient Method
10.2.5 Conjugate Direction Method
10.2.6 Other Derivative-Free Methods
10.3 Computational Methods in Nonlinear Regression
10.3.1 Newton’s Method for the Nonlinear Regression Problem
10.3.2 The Modified Gauss-Newton Method
10.3.3 The Levenberg-Marquardt Modification of Gauss-Newton
10.3.4 Alternative Gradient Methods
10.3.5 Minimization Without Derivatives
10.3.6 Summary
10.4 Test Problems
Exercises
References and Further Readings
11/ Model Fitting Based on Criteria Other Than Least Squares
11.1 Introduction
11.2 Minimum L Norm Estimators
11.2.1 Lᵖ Estimation
11.2.2 L I͚ Estimators
11.4 Biased Estimation
11.5 Robust Nonlinear Regression
Exercises
References and Further Readings
12/ Selected Multivariate Methods
12.1 Introduction
12.2 Canonical Correlations
12.3 Principal Components
12.4 Factor Analysis
12.5 Multivariate Analysis of Variance
Exercises
References and Further Readings
Index

Biography

Martorell, Sebastián; Guedes Soares, Carlos; Barnett, Julie

"The publication of this book, I believe, is a milestone. . .Kennedy and Gentle have done an outstanding job of assembling the best techniques from a great variety of sources, establishing a benchmark for the field of statistical computing. "
---Mathematics of Computation
". . .a very impressive text. . .highly readable and well illustrated with examples. . . .the reader who intends to take a hand in designing his own regression and multivariate packages will find a storehouse of information. "
---Journal of the American Statistical Association
". . .a valuable addition to the literature on statistical computing. "
---Mathematical Reviews