Multivariate density estimation Elektronische Ressource theory, practice, and visualization David W. Scott, Rice University Houston, Texas

Includes bibliographical references and index

Title Page; Copyright Page; Contents; Preface to Second Edition; Preface to First Edition; Chapter 1 Representation and Geometry of Multivariate Data; 1.1 Introduction; 1.2 Historical Perspective; 1.3 Graphical Display of Multivariate Data Points; 1.3.1 Multivariate Scatter Diagrams; 1.3.2 Chernoff Faces; 1.3.3 Andrews' Curves and Parallel Coordinate Curves; 1.3.4 Limitations; 1.4 Graphical Display of Multivariate Functionals; 1.4.1 Scatterplot Smoothing by Density Function; 1.4.2 Scatterplot Smoothing by Regression Function; 1.4.3 Visualization of Multivariate Functions.

1.4.3.1 Visualizing Multivariate Regression Functions1.4.4 Overview of Contouring and Surface Display; 1.5 Geometry of Higher Dimensions; 1.5.1 Polar Coordinates in d Dimensions; 1.5.2 Content of Hypersphere; 1.5.3 Some Interesting Consequences; 1.5.3.1 Sphere Inscribed in Hypercube; 1.5.3.2 Hypervolume of a Thin Shell; 1.5.3.3 Tail Probabilities of Multivariate Normal; 1.5.3.4 Diagonals in Hyperspace; 1.5.3.5 Data Aggregate Around Shell; 1.5.3.6 Nearest Neighbor Distances; Problems; Chapter 2 Nonparametric Estimation Criteria; 2.1 Estimation of the Cumulative Distribution Function.

2.2 Direct Nonparametric Estimation of the Density2.3 Error Criteria for Density Estimates; 2.3.1 MISE for Parametric Estimators; 2.3.1.1 Uniform Density Example; 2.3.1.2 General Parametric MISE Method with Gaussian Application; 2.3.2 The L1 Criterion; 2.3.2.1 L1 versus L2; 2.3.2.2 Three Useful Properties of the L1 Criterion; 2.3.3 Data-Based Parametric Estimation Criteria; 2.4 Nonparametric Families of Distributions; 2.4.1 Pearson Family of Distributions; 2.4.2 When Is an Estimator Nonparametric?; Problems; Chapter 3 Histograms: Theory and Practice.

3.1 Sturges' Rule for Histogram Bin-Width Selection3.2 The L2 Theory of Univariate Histograms; 3.2.1 Pointwise Mean Squared Error and Consistency; 3.2.2 Global L2 Histogram Error; 3.2.3 Normal Density Reference Rule; 3.2.3.1 Comparison of Bandwidth Rules; 3.2.3.2 Adjustments for Skewness and Kurtosis; 3.2.4 Equivalent Sample Sizes; 3.2.5 Sensitivity of MISE to Bin Width; 3.2.5.1 Asymptotic Case; 3.2.5.2 Large-Sample and Small-Sample Simulations; 3.2.6 Exact MISE versus Asymptotic MISE; 3.2.6.1 Normal Density; 3.2.6.2 Lognormal Density; 3.2.7 Influence of Bin Edge Location on MISE.

3.2.7.1 General Case3.2.7.2 Boundary Discontinuities in the Density; 3.2.8 Optimally Adaptive Histogram Meshes; 3.2.8.1 Bounds on MISE Improvement for Adaptive Histograms; 3.2.8.2 Some Optimal Meshes; 3.2.8.3 Null Space of Adaptive Densities; 3.2.8.4 Percentile Meshes or Adaptive Histograms with Equal Bin Counts; 3.2.8.5 Using Adaptive Meshes versus Transformation; 3.2.8.6 Remarks; 3.3 Practical Data-Based Bin Width Rules; 3.3.1 Oversmoothed Bin Widths; 3.3.1.1 Lower Bounds on the Number of Bins; 3.3.1.2 Upper Bounds on Bin Widths; 3.3.2 Biased and Unbiased CV; 3.3.2.1 Biased CV.

3.3.2.2 Unbiased CV.

Clarifies modern data analysis through nonparametric density estimation for a complete working knowledge of the theory and methods Featuring a thoroughly revised presentation, Multivariate Density Estimation: Theory, Practice, and Visualization, Second Edition maintains an intuitive approach to the underlying methodology and supporting theory of density estimation. Including new material and updated research in each chapter, the Second Edition presents additional clarification of theoretical opportunities, new algorithms, and up-to-date coverage of the unique challenges presented in the fie

Clarifies modern data analysis through nonparametric density estimation for a complete working knowledge of the theory and methods Featuring a thoroughly revised presentation, Multivariate Density Estimation: Theory, Practice, and Visualization, Second Edition maintains an intuitive approach to the underlying methodology and supporting theory of density estimation. Including new material and updated research in each chapter, the Second Edition presents additional clarification of theoretical opportunities, new algorithms, and up-to-date coverage of the unique challenges presented in the field of data analysis. The new edition focuses on the various density estimation techniques and methods that can be used in the field of big data. Defining optimal nonparametric estimators, the Second Edition demonstrates the density estimation tools to use when dealing with various multivariate structures in univariate, bivariate, trivariate, and quadrivariate data analysis. Continuing to illustrate the major concepts in the context of the classical histogram, Multivariate Density Estimation: Theory, Practice, and Visualization, Second Edition also features: -Over 150 updated figures to clarify theoretical results and to show analyses of real data sets -An updated presentation of graphic visualization using computer software such as R -A clear discussion of selections of important research during the past decade, including mixture estimation, robust parametric modeling algorithms, and clustering -More than 130 problems to help readers reinforce the main concepts and ideas presented -Boxed theorems and results allowing easy identification of crucial ideas -Figures in color in the digital versions of the book -A website with related data sets Multivariate Density Estimation: Theory, Practice, and Visualization, Second Edition is an ideal reference for theoretical and applied statisticians, practicing engineers, as well as readers interested in the theoretical aspects of nonparametric estimation and the application of these methods to multivariate data. The Second Edition is also useful as a textbook for introductory courses in kernel statistics, smoothing, advanced computational statistics, and general forms of statistical distributions