The First Detailed Account of Statistical Analysis That Treats Models as Approximations The idea of truth plays a role in both Bayesian and frequentist statistics. The Bayesian concept of coherence is based on the fact that two different models or parameter values cannot both be true. Frequentist statistics is formulated as the problem of estimating the "true but unknown" parameter value that generated the data. Forgoing any concept of truth, Data Analysis and Approximate Models: Model Choice, Location-Scale, Analysis of Variance, Nonparametric Regression and Image Analysis presents statistical analysis/inference based on approximate models. Developed by the author, this approach consistently treats models as approximations to data, not to some underlying truth. The author develops a concept of approximation for probability models with applications to: Discrete data Location scale Analysis of variance (ANOVA) Nonparametric regression, image analysis, and densities Time series Model choice The book first highlights problems with concepts such as likelihood and efficiency and covers the definition of approximation and its consequences. A chapter on discrete data then presents the total variation metric as well as the Kullback-Leibler and chi-squared discrepancies as measures of fit. After focusing on outliers, the book discusses the location-scale problem, including approximation intervals, and gives a new treatment of higher-way ANOVA. The next several chapters describe novel procedures of nonparametric regression based on approximation. The final chapter assesses a range of statistical topics, from the likelihood principle to asymptotics and model choice.
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
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