• Multiple Analyses and Multiple Views
• Simple Plots May Reveal the Unexpected

• Descriptive Statistics, Inferential Statistics, and Model Building

• Statistical Functions of the CDF and the ECDF
• Estimation of Statistical Functions
• Estimation Using the ECDF
• Empirical Quantiles

• Some Comments on Optimization
• Estimation by Minimizing Residuals
• Statistical Properties of Minimum-Residual Estimators
• Least Squares Estimation
• Variance of Least Squares Estimators
• Iteratively Reweighted Least Squares
• Estimation by Maximum Likelihood
• Statistical Properties of MLE
• EM Methods

• Functions of Parameters and Functions of Estimators
• Linear Estimators

• Tests of Hypotheses
• Confidence Intervals

2 Monte Carlo Methods for Statistical Inference ... 39

• Inverse CDF Method
• Acceptance/Rejection Methods
• Use of Conditional Distributions
• Acceptance/Rejection Method Using a Markov Chain
• Generation of Multivariate Random Variates
• Gibbs Sampling
• Probability Densities Known Only Proportionally
• Data-Based Random Number Generation

• Estimation of a Definite Integral
• Estimation of the Variance
• Estimating the Variance Using Batch Means
• Convergence of Iterative Monte Carlo and Mixing of the Markov Chain
• Monte Carlo, Iterative Monte Carlo, and Simulation

• Importance Sampling
• Control Variates

3 Randomization and Data Partitioning ... 69

• Jackknife Variance Estimate
• Jackknife Bias Correction
• Higher-Order Bias Corrections
• The Generalized Jackknife
• The Delete-$k$ Jackknife

4 Bootstrap Methods ... 85

• Basic Intervals
• Bootstrap-$t$ Intervals
• Bootstrap Percentile Confidence Intervals
• Confidence Intervals Based on Transformations
• Correcting the Bias in Intervals Due to Bias in the Estimator or to Lack of Symmetry

• Jackknife After Bootstrap
• The Bootstrap Estimate of the Bias of a Plug-In Estimator
• Balanced Resampling

5 Tools for Identification of Structure in Data ... 99

• Flats ... 101

• Orthogonal Transformations
• Gram-Schmidt Orthogonalization
• Geometric Transformations
• Rotations
• Projections

• Similarities: Covariances and Correlations
• Similarities When Some Variables Are Categorical
• Similarities among Functional Observations
• Similarities between Groups of Variables
• Dissimilarities: Distances
• Other Dissimilarities Based on Distances
• Dissimilarities in Anisometric Coordinate Systems: Sphering Data
• Properties of Dissimilarities
• Dissimilarities between Groups of Observations
• Effects of Transformations of the Data
• Outlying Observations and Collinear Variables
• Multidimensional Scaling: Determining Observations that Yield a Given Distance Matrix

6 Estimation of Functions ... 127

• Inner Products and Norms
• Complete Spaces
• Measures for Comparing Two Functions
• Basis Sets in Function Spaces
• Series Expansions in Basis Functions
• Orthogonal Polynomials
• Multivariate Orthogonal Polynomials
• Function Decomposition and Estimation of the Coefficients in an Orthogonal Expansion
• Splines
• Interpolating Splines
• Smoothing Splines
• Choice of Knots in Smoothing Splines
• Multivariate Splines
• Kernel Methods

• Bias
• Variance
• Mean Squared Error
• Mean Absolute Error
• Consistency

• Integrated Bias and Variance
• Integrated Mean Squared Error and Mean Absolute Error
• Mean SAE
• Large-Sample Statistical Properties
• Other Global Properties of Estimators of Functions

7 Graphical Methods in Computational Statistics ... 153

• Histograms and Variations
• The Empirical Cumulative Distribution Function and q-q Plots
• Smoothing
• Graphing Continuous Functions
• B\'{e}zier Curves
• Continuous Densities
• Representation of the Third Dimension
• Contours in Three Dimensions

• Projections
• Conditioning Plots
• Noncartesian Displays
• Glyphs and Icons
• Parallel Coordinates: Points Become Broken Line Segments
• Trigonometric Series: Points Become Curves
• Cone Plots
• Exploring Data with Noncartesian Displays
• Roping, Brushing, and Linking
• Rotations and Dynamical Graphics
• Displays of Large Data Sets
• Data Analysis and Human Perception

• Speed and Resolution
• Representation of Color
• Low-Level Software

Part II. Exploring Data Density and Structure

8 Estimation of Probability Density Functions Using Parametric Models ... 197

• Maximum Likelihood Methods
• Fitting by Matching Moments
• Fitting by Matching Quantiles
• Statistical Properties of Parametric Family Estimators ... 199

• Pearson Curves
• Other General Parametric Univariate Families of Distribution Functions

9 Nonparametric Estimation of Probability Density Functions ... 205

• Some Properties of the Histogram Estimator
• Pointwise and Binwise Properties
• Asymptotic MISE (or AMISE) of Histogram Estimators
• Bin Sizes
• Bin Shapes
• Other Density Estimators Related to the Histogram

• Rosenblatt's Histogram Estimator; Kernels
• Properties of Kernel Estimators
• Choice of Kernels
• Computation of Kernel Density Estimators

• Filtered Kernel Methods
• Alternating Kernel and Mixture Methods
• Methods of Comparisons of Methods

10 Structure in Data ... 233

• Clustering
• K-Means Clustering
• Choosing the Number of Clusters
• Hierarchical Clustering
• Agglomerative Hierarchical Clustering
• Model-Based Hierarchical Clustering
• Divisive Hierarchical Clustering
• Other Divisive Clustering Schemes
• Clustering and Classification by Space Tessellations
• Meanings of Clusters; Conceptual Clustering
• Fuzzy Clustering
• Clustering and Transformations of the Data
• Clustering of Variables
• Comparing Clusterings
• Computational Complexity of Clustering

• Minimal Spanning Trees
• Ranking Data Using Minimal Spanning Trees
• Ranking Data Using Convex Container Hulls
• Ranking Data Using Location Depth
• Ordering by Clustering
• Clustering by Ordering
• Nearest Neighbors and $k$-$d$-Trees
• Ordering and Ranking of Transformed Data

• The Probability Model Underlying Principal Components Analysis
• Principal Components Analysis of Data
• Dimension Reduction by Principal Components Analysis
• Principal Components and Transformations of the Data
• Principal Components of Observations
• Principal Components Directly from the Data Matrix
• Computational Issues
• PCA for Clustering
• Robustness of Principal Components

• Factor Analysis
• The Probability Model Underlying Factor Analysis
• Factor Analysis of Data
• Latent Semantic Indexing
• Linear Independent Components Analysis

• The Probability Model Underlying Projection Pursuit
• Projection Indexes for the Probability Model
• Projection Pursuit in Data
• Exploratory Projection Pursuit
• Example
• Computational Issues

• Independent Components Analysis

• Data Sparsity in Higher Dimensions
• Volumes of Hyperspheres and Hypercubes
• The Curse of Dimensionality
• Tiling Space

11 Statistical Models of Dependencies ... 299

• Generalized Models
• Classification Models
• Models of Sequential Dependencies
• Data and Models
• Transformations
• Piecewise Models

• Hierarchical Models
• Probability Distributions in Models of Sequential Dependencies

• The Mechanics of Fitting
• Estimation by Minimizing Residuals
• Variations on Minimizing Residuals
• Comparisons of Estimators Defined by Minimum Residuals
• Orthogonal Residuals
• Projection Pursuit Regression
• Classification and Regression Trees
• Combining Classifications
• Kernel Methods and Support Vector Machines
• Fitting Models of Sequential Dependencies
• Transformations to Make Data Fit Models
• Variance Stabilization
• Transformations of the Independent Variables
• Transformations of the Box-Cox Type
• Alternating Conditional Expectation
• Additivity and Variance Stabilization
• Local Fitting
• Assessing the Fit of a Model

Appendices

A Monte Carlo Studies in Statistics ... 337

• The Problem
• The Design of the Experiment
• The Experiment
• Reporting the Results

• Latin Hypercube Sampling

B Software for Random Number Generation ... 351

• Controlling the State of the Generators

• Controlling the State of the Generators
• Monte Carlo in S-Plus and R

C Notation and Definitions ... 363

D Solutions and Hints for Selected Exercises ... 377

Bibliography ... 385

Author Index ... 409

Subject Index ... 415