Elements of Computational Statistics

by Gentle, James E.

• ISBN13: 9780387954899
• ISBN10: 0387954899
• Format: Hardcover
• Copyright: 2002-12-01
• Publisher: Springer Verlag

Summary

Computationally intensive methods have become widely used both for statistical inference and for exploratory analyses of data. The methods of computational statistics involve resampling, partitioning, and multiple transformations of a dataset. They may also make use of randomly generated artificial data. Implementation of these methods often requires advanced techniques in numerical analysis, so there is a close connection between computational statistics and statistical computing. This book describes techniques used in computational statistics, and addresses some areas of application of computationally intensive methods, such as density estimation, identification of structure in data, and model building. Although methods of statistical computing are not emphasized in this book, numerical techniques for transformations, for function approximation, and for optimization are explained in the context of the statistical methods. The book includes exercises, some with solutions. The book can be used as a text or supplementary text for various courses in modern statistics at the advanced undergraduate or graduate level, and it can also be used as a reference for statisticians who use computationally-intensive methods of analysis. Although some familiarity with probability and statistics is assumed, the book reviews basic methods of inference, and so is largely self-contained. James Gentle is University Professor of Computational Statistics at George Mason University. He is a Fellow of the American Statistical Association and a member of the International Statistical Institute. He has held several national offices in the American Statistical Association and has served as associate editor for journals of the ASA as well as for other journals in statistics and computing. He is the author of Random Number Generation and Monte Carlo Methods and Numerical Linear Algebra for Statistical Applications.

Table of Contents

Preface	vii
I Methods of Computational Statistics	1	(190)
Introduction to Part I	3	(2)
Preliminaries	5	(34)
Discovering Structure: Data Structures and Structure in Data	6	(2)
Modeling and Computational Inference	8	(3)
The Role of the Empirical Cumulative Distribution Function	11	(4)
The Role of Optimization in Inference	15	(15)
Inference about Functions	30	(2)
Probability Statements in Statistical Inference	32	(7)
Exercises	35	(4)
Monte Carlo Methods for Statistical Inference	39	(30)
Generation of Random Numbers	40	(13)
Monte Carlo Estimation	53	(5)
Simulation of Data from a Hypothesized Model: Monte Carlo Tests	58	(2)
Simulation of Data from a Fitted Model: ``Parametric Bootstraps''	60	(1)
Random Sampling from Data	60	(1)
Reducing Variance in Monte Carlo Methods	61	(4)
Acceleration of Markov Chain Monte Carlo Methods	65	(4)
Exercises	66	(3)
Randomization and Data Partitioning	69	(16)
Randomization Methods	70	(4)
Cross Validation for Smoothing and Fitting	74	(2)
Jackknife Methods	76	(9)
Further Reading	82	(1)
Exercises	83	(2)
Bootstrap Methods	85	(14)
Bootstrap Bias Corrections	86	(2)
Bootstrap Estimation of Variance	88	(1)
Bootstrap Confidence Intervals	89	(4)
Bootstrapping Data with Dependencies	93	(1)
Variance Reduction in Monte Carlo Bootstrap	94	(5)
Further Reading	96	(1)
Exercises	97	(2)
Tools for Identification of Structure in Data	99	(28)
Linear Structure and Other Geometric Properties	100	(1)
Linear Transformations	101	(7)
General Transformations of the Coordinate System	108	(1)
Measures of Similarity and Dissimilarity	109	(14)
Data Mining	123	(1)
Computational Feasibility	124	(3)
Exercises	125	(2)
Estimation of Functions	127	(26)
General Methods for Estimating Functions	128	(15)
Pointwise Properties of Function Estimators	143	(3)
Global Properties of Estimators of Functions	146	(7)
Exercises	150	(3)
Graphical Methods in Computational Statistics	153	(38)
Viewing One, Two, or Three Variables	155	(13)
Viewing Multivariate Data	168	(16)
Hardware and Low-Level Software for Graphics	184	(2)
Software for Graphics Applications	186	(5)
Further Reading	188	(1)
Exercises	188	(3)
II Exploring Data Density and Structure	191	(145)
Introduction to Part II	193	(4)
Estimation of Probability Density Functions Using Parametric Models	197	(8)
Fitting a Parametric Probability Distribution	198	(1)
General Families of Probability Distributions	199	(3)
Mixtures of Parametric Families	202	(3)
Exercises	203	(2)
Nonparametric Estimation of Probability Density Functions	205	(28)
The Likelihood Function	206	(2)
Histogram Estimators	208	(9)
Kernel Estimators	217	(5)
Choice of Window Widths	222	(1)
Orthogonal Series Estimators	222	(2)
Other Methods of Density Estimation	224	(9)
Exercises	226	(7)
Structure in Data	233	(66)
Clustering and Classification	237	(18)
Ordering and Ranking Multivariate Data	255	(9)
Linear Principal Components	264	(12)
Variants of Principal Components	276	(5)
Projection Pursuit	281	(8)
Other Methods for Identifying Structure	289	(1)
Higher Dimensions	290	(9)
Exercises	294	(5)
Statistical Models of Dependencies	299	(37)
Regression and Classification Models	301	(7)
Probability Distributions in Models	308	(3)
Fitting Models to Data	311	(25)
Exercises	333	(3)
Appendices	336	(49)
A Monte Carlo Studies in Statistics	337	(14)
A.1 Simulation as an Experiment	338	(1)
A.2 Reporting Simulation Experiments	339	(1)
A.3 An Example	340	(7)
A.4 Computer Experiments	347	(2)
Exercises	349	(2)
B Software for Random Number Generation	351	(12)
B.1 The User Interface for Random Number Generators	353	(1)
B.2 Controlling the Seeds in Monte Carlo Studies	354	(1)
B.3 Random Number Generation in IMSL Libraries	354	(3)
B.4 Random Number Generation in S-Plus and R	357	(6)
C Notation and Definitions	363	(14)
D Solutions and Hints for Selected Exercises	377	(8)
Bibliography	385	(24)
Literature in Computational Statistics	386	(1)
Resources Available over the Internet	387	(2)
References for Software Packages	389	(1)
References to the Literature	389	(20)
Author Index	409	(6)
Subject Index	415