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Robustness in Data Analysis: Criteria and Methods

By Georgy L. Shevlyakov and Nikita O. Vilchevski
December 2001
VSP
ISBN: 90-6764-351-3
324 pages
$187.50 hardcover

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The field of mathematical statistics called robustness statistics deals with the stability of statistical inference under variations of accepted distribution models. Although robust statistics involves mathematically highly defined tools, robust methods exhibit a satisfactory behaviour in small samples, thus being quite useful in applications.

This volume in the book series Modern Probability and Statistics addresses various topics in the field of robust statistics and data analysis, such as: a probability-free approach in data analysis; minimax variance estimators of location, scale, regression, autoregression and correlation; L1-norm methods; adaptive, data reduction, bivariate boxplot, and multivariate outlier detection algorithms; applications in reliability, detection of signals, and analysis of the sudden cardiac death risk factors.

The book contains new results related to robustness and data analysis technologies, including both theoretical aspects and practical needs of data processing, which have been relatively inaccessible as they were originally only published in Russian.

This book will be of value and interest to researchers in mathematical statistics as well as to those using statistical methods.

Contents:

INTRODUCTION

General remarks

Huber minimax approach

Hampel approach

OPTIMIZATION CRITERIA IN DATA ANALYSIS: A PROBABILITY-FREE APPROACH

Introductory remarks

Translation and scale equivariant contrast functions

Orthogonal equivariant contrast functions

Monotonically equivariant contrast functions

Minimal sensitivity to small perturbations in the data

Affine equivariate contrast functions

ROBUST MIMIMAX ESTIMATION OF LOCATION

Introductory remarks

Robust estimation of location in models with bounded variances

Robust estimation of location in models with bounded subranges

Robust estimators of multivariate location

Least informative lattice distributions

ROBUST ESTIMATION OF SCALE

Introductory remarks; Measures of scale defined by functionals

M-, L-, and R-estimators of scale

Huber minimax estimator of scale

Final remarks

ROBUST REGRESSION AND AUTOREGRESSION

Introductory remarks

The minimax variance regression

Robust autoregression

Robust identification in dynamic models

Final remarks

ROBUSTNESS OF L₁-NORM ESTIMATORS

Introductory remarks

Stability of L₁-approximations

Robustness of the L₁-regression

Final remarks

ROBUST ESTIMATION OF CORRELATION

Introductory remarks

Analysis: Monte Carlo experiment

Analysis: asymptotic characteristics

Synthesis: minimax variance correlation

Two-stage estimators: rejection of outliers plus classics

COMPUTATION AND DATA ANALYSIS TECHNOLOGIES

Introductory remarks on computation

Adaptive robust procedures

Smoothing quantile functions by the Bernstein polynomials

Robust bivariate boxplots

APPLICATIONS

On robust elimination in the statistical theory of reliability

Robust detection of signals based on optimisation criteria

Statistical analysis of sudden cardiac death risk factors

Bibliography

Index

Mathematics
Modern Probability & Statistics Series