**Also available as eBook on:**

In this book the authors have assembled the best techniques from a great variety of sources, establishing a benchmark for the field of statistical computing. ---Mathematics of Computation . The text is 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 and a valuable resource in the field of 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