Statistical Computing

 
 
Taylor & Francis Ltd. (Verlag)
  • erschienen am 14. Dezember 2018
  • |
  • 608 Seiten
 
E-Book | PDF mit Adobe DRM | Systemvoraussetzungen
978-1-351-41459-3 (ISBN)
 
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.
  • Englisch
  • Boca Raton
  • |
  • USA
978-1-351-41459-3 (9781351414593)
weitere Ausgaben werden ermittelt
  • Cover
  • Half Title
  • Title Page
  • Copyright Page
  • Contents
  • Preface
  • 1 INTRODUCTION
  • 1.1 Orientation
  • 1.2 Purpose
  • 1.3 Prerequisites
  • 1.4 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 POINTS IN 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 Distribution
  • 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 Based 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 COHPUTATIONAL 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 Methodsfor 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 NonlinearRegression 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 Lp Norm Estimators
  • 11.2.1 L1 Estimation
  • 11.2.2 L[omited] Estimation
  • 11.2.3 Other Lp Estimators
  • 11.3 Other Robust 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

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