1st Edition

Data Analysis and Approximate Models Model Choice, Location-Scale, Analysis of Variance, Nonparametric Regression and Image Analysis

By Patrick Laurie Davies Copyright 2014
    320 Pages 110 B/W Illustrations
    by Chapman & Hall

    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.

    Approximate Models
    Two Modes of Statistical Analysis
    Towards One Mode of Analysis
    Approximation, Randomness, Chaos, Determinism

    A Concept of 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
    Stability of Analysis
    The Choice of En(α, P)
    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(λ) Model
    The b(k, p) and nb(k, p) Models
    The Flying Bomb Data
    The Student Study Times Data

    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
    Efficiency and Regularization
    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

    Nonparametric Regression: Location
    A Definition of Approximation
    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
    Approximation Regions and Regularization
    The Taut String Strategy for Densities
    Smoothing the Taut String Approximation

    A Critique of Statistics
    Bayesian Statistics
    Sufficient Statistics
    Model Choice
    What Can Actually Be Estimated?




    Patrick Laurie Davies

    "Davies tackles the problem of the foundations of statistics. … reading this book will make you think and question your own views on statistics. It reminds us that the foundations of statistics are still, and more than ever, open to discussion."
    Mathematical Reviews, August 2015