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

320 Pages
by Chapman & Hall

Also available as eBook on:

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(α, 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(λ) 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

Biography

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