# 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.
 Reihe: Sprache: Englisch Verlagsort: Boca Raton | USA Schlagworte: ISBN-13: 978-1-351-41459-3 (9781351414593)
weitere Ausgaben werden ermittelt
• Cover
• Half Title
• Title 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
• 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
• 5.3 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
• 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
• 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
• 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
• 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
• 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
• 10.2.5 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
• 10.3.5 Minimization Without Derivatives
• 10.3.6 Summary
• 10.4 Test Problems
• Exercises
• 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
• 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
• Index
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