Data Analysis and Approximate Models: Model Choice, Location-Scale, Analysis of Variance, Nonparametric Regression and Image Analysis, 1st Edition (Hardback) book cover

Data Analysis and Approximate Models

Model Choice, Location-Scale, Analysis of Variance, Nonparametric Regression and Image Analysis, 1st Edition

By Patrick Laurie Davies

Chapman and Hall/CRC

320 pages | 110 B/W Illus.

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pub: 2014-07-07
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Description

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.

Reviews

"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

Table of Contents

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

About the Series

Chapman & Hall/CRC Monographs on Statistics and Applied Probability

Learn more…

Subject Categories

BISAC Subject Codes/Headings:
MAT029000
MATHEMATICS / Probability & Statistics / General
MAT029010
MATHEMATICS / Probability & Statistics / Bayesian Analysis