Data analysis and approximate models : model choice, location-scale, analysis of variance, nonparametic regression and image analysis

 Content: Introduction  Introduction Approximate Models  Notation  Two Modes of Statistical Analysis Towards One Mode of Analysis  Approximation, Randomness, Chaos, Determinism  Approximation  A Concept of Approximation  Approximation  Approximating a Data Set by a Model  Approximation Regions  Functionals and Equivariance Regularization and Optimality  Metrics and Discrepancies Strong and Weak Topologies  On Being (almost) Honest  Simulations and Tables  Degree of Approximation and p-values  Scales Stability of Analysis  The Choice of En(alpha, P)  Independence  Procedures, Approximation and Vagueness  Discrete Models  The Empirical Density  Metrics and Discrepancies  The Total Variation Metric  The Kullback-Leibler and Chi-Squared Discrepancies  The Po(lambda) Model The b(k, p) and nb(k, p) Models  The Flying Bomb Data  The Student Study Times Data   Outliers Outliers, Data Analysis and Models  Breakdown Points and Equivariance  Identifying Outliers and Breakdown  Outliers in Multivariate Data  Outliers in Linear Regression  Outliers in Structured Data   The Location-Scale Problem Robustness  Efficiency and Regularization  M-functionals Approximation Intervals, Quantiles and Bootstrapping  Stigler's Comparison of Eleven Location Functionals Based on Historical Data Sets  An Attempt at an Automatic Procedure  Multidimensional M-functionals   The Analysis of Variance  The One-Way Table The Two-Way Table The Three-Way and Higher Tables Interactions in the Presence of Noise Examples   Nonparametric Regression: Location  A Definition of Approximation  Regularization Rates of Convergence and Approximation Bands  Choosing Smoothing Parameters Joint Approximation of Two or More Samples Inverse Problems  Heterogeneous Noise   Nonparametric Regression: Scale  The Standard Model and a Concept of Approximation  Piecewise Constant Scale and Local Approximation  GARCH Segmentation  The Taut String and Scale  Smooth Scale Functions  Comparison of the Four Methods  Location and Scale  Image Analysis  Two and Higher Dimensions  The Approximation Region Linear Programming and Related Methods  Choosing Smoothing Parameters  Nonparametric Densities  Introduction  Approximation Regions and Regularization The Taut String Strategy for Densities  Smoothing the Taut String Approximation  A Critique of Statistics  Likelihood  Bayesian Statistics  Sufficient Statistics  Efficiency  Asymptotics  Model Choice  What Can Actually Be Estimated?   Bibliography Index
 Abstract:             "This book presents a philosophical study of statistics via the concept of data approximation. Developed by the well-regarded author, this approach discusses how analysis must take into account that models are, at best, an approximation of real data. It is, therefore, closely related to robust statistics and nonparametric statistics and can be used to study nearly any statistical technique. The book also includes an interesting discussion of the frequentist versus Bayesian debate in statistics. "--
"This book stems from a dissatisfaction with what is called formal statistical inference. The dissatisfaction started with my first contact with statistics in a course of lectures given by John Kingman in Cambridge in 1963. In spite of Kingman's excellent pedagogical capabilities it was the only course in the Mathematical Tripos I did not understand. Kingman later told me that the course was based on notes by Dennis Lindley, but the approach given was not a Bayesian one. From Cambridge I went to LSE where I did an M.Sc. course in statistics. Again, in spite of excellent teachers including David Brillinger, Jim Durbin and Alan Stuart I did not really understand what was going on. This did not prevent me from doing whatever I was doing with success and I was awarded a distinction in the final examinations. Later I found out that I was not the only person who had problems with statistics. Some years ago I asked a respected German colleague D.W. Müller of the University of Heidelberg why he had chosen statistics. He replied that it was the only subject he had not understood as a student. Frank Hampel has even written an article entitled 'Is statistics too difficult?'. I continued at LSE and wrote my Ph. D. thesis on random entire functions under the supervision of Cyril Offord. It involved no statistics whatsoever. From London I moved to Constance in Germany, from there to Sheffield, then back to Germany to the town of Münster. All the time I continued writing papers in probability theory including some on the continuity properties of Gaussian processes. At that time Jack Cuzick now of Queen Mary, University of London, and Cancer Research UK also worked on this somewhat esoteric subject."