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Robust statistical methods with R. (English) Zbl 1097.62020
Boca Raton, FL: Chapman and Hall/CRC (ISBN 1-58488-454-1/hbk). xi, 197 p. (2006).
Robust statistical methods were developed to supplement classical procedures when the data violate classical assumptions. They are ideally suited to applied research across a broad spectrum of study. This book has intended to provide a systematic treatment of robust procedures with an emphasis on practical applications in a limited space (less than 200 pages). The authors work from underlying mathematical tools to implementation paying special attention to the computational aspects. They cover many robust methods, including differentiable statistical functions, distance of measures, influence functions, and asymptotic distributions. Highlighting hands-on problem solving, examples and computational algorithms using the R software supplement the discussion. The book also examines the characteristics of robustness, estimators of real parameters, some large sample properties, and several goodness-of-fit tests. It also includes a brief overview of R in an appendix for those with litter experience using the software. The book is organized as follows:
In Chapter 1, the mathematical tools of robustness are briefly discussed, which includes the introduction to statistical models, estimation of parameters, statistical functionals, Fisher consistency, distances of probability measures, derivatives of statistical functionals, and large sample distributions of empirical functionals. Chapter 2 introduces the basic characteristics of robustness with brief discussion including influence functions, qualitative robustness, maximum bias, breakdown point, and tail-behavior of estimators. In Chapter 3, robust estimators of a real parameter are studied with emphasis on M-estimators, L-estimators and R-estimators and their properties. Robust estimators in linear models are considered in Chapter 4, where least squares method, M-estimators, GM-estimators, S-estimators, MM-estimators, L-estimators, regression quantiles, regression rank scores, robust scale statistics, estimators with high breakdown points, and one-step versions of estimators are covered. Chapter 5 is devoted to the robust estimation of location and scatter in multivariate location models. Some large sample properties of robust procedures are given in Chapter 6. Several goodness-of-fit tests are presented in Chapter 7. A brief R overview is given in the appendix.
In this book, problems and complements are provided in the end of most of chapters. Computation and software notes can also be found in some chapters. This book is intended to serve as a text for graduate and post-graduate studies as well as a reference for statisticians and quantitative scientists.

MSC:
62F35 Robustness and adaptive procedures (parametric inference)
62-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to statistics
65C60 Computational problems in statistics (MSC2010)
62G35 Nonparametric robustness
62-04 Software, source code, etc. for problems pertaining to statistics
Keywords:
Admissibility; Andrews sinus function; asymptotic distribution; asymptotic relative efficiency; asymptotic representation; asymptotic variance; asymptotic equivalent estimator; balance design; breakdown point; likelihood; contaminated distribution; BLUE; Dirac probability measure; distance of measures; Kullback-Leibler divergence; Lévy distance; Prochorov distance; Hellinger distance; Kolmogorov distance; Lipschitz distance; total variation; empirical distribution; quantile; equivariance; exponential tail; minimax; Fisher consistency; Fisher information; global sensitivity; good-of-fit test; Gini mean difference; Shapiro-Wilks test; heavy tailed distribution; Huber estimator; Huber function; Tukey biweight function; Hampel \(\psi\) function; skipped mean; skipped median; influence function; inter-quantile range; James-Stein estimator; \(L_1\)-multivariate median; least favorable distribution; least median of squares; least squares; least trimmed squares estimator; least trimmed sum of absolute deviation; L-estimator; leverage point; local sensitivity; likelihood ratio; maximum bias; median; median absolute deviation; median unbiased estimator; M-estimator; GM-estimator; asymptotic risk-efficiency; M-functional; midrange; minimax robustness; minimum covariance determinant estimator; minimum volume ellipsoid estimator; MM-estimator; moment estimator; multivariate t-distribution; multivariate normal distribution; one-step version; Pitman closeness; pivot; regression quantile; qualitative robustness; R system; Rao-Cramér lower bound; redescending function; regression equivalence; scale equivalence; regression invariance; regression rank scores; R-estimator; robust estimator; scale statistic; scatter matrix; Sen’s weighted mean; sequential estimator; shrinkage phenomenon; t-test; F-test; S-estimator; spatial rank function; spherically symmetric distribution; Fréchet derivative; Gateau derivative; Hadamard derivative; differentiability; tail behavior measure; statistical model; Studentized M-estimator; translation equivalence; Winsorized mean
Software:
ROBETH; R
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