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Another approach to asymptotics and bootstrap of randomly trimmed means
 Annals of the Institute of Statistical Mathematics
, 2004
"... A unified, empirical processes based approach to the central limit theorem and to the bootstrap for randomly trimmed and Winsorized means is developped, with emphasis on Hampel’s means. ..."
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A unified, empirical processes based approach to the central limit theorem and to the bootstrap for randomly trimmed and Winsorized means is developped, with emphasis on Hampel’s means.
Approximating data and statistical procedures. I. Approximating data
, 2003
"... Stochastic models approximate data and are not true representations of the same. Statistical procedures make use of approximate stochastic models to facilitate the analysis of data. ..."
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Stochastic models approximate data and are not true representations of the same. Statistical procedures make use of approximate stochastic models to facilitate the analysis of data.
Statistical procedures and robust statistics
 Estadistica
"... It is argued that a main aim of statistics is to produce statistical procedures which in this article are defined as algorithms with inputs and outputs. The structure and properties of such procedures are investigated with special reference to topological and testing considerations. Procedures whi ..."
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It is argued that a main aim of statistics is to produce statistical procedures which in this article are defined as algorithms with inputs and outputs. The structure and properties of such procedures are investigated with special reference to topological and testing considerations. Procedures which work well in a large variety of situations are often based on robust statistical functionals. In the final section some aspects of robust statistics are discussed again with special reference to topology and continuity. 1
Uniform asymptotics for S and MMregression estimators
, 2006
"... In this paper we find verifiable regularity conditions to ensure that Sestimators of scale and regression and MMestimators of regression are uniformly consistent and uniformly asymptotically normally distributed over contamination neighbourhoods. Moreover, we show how to calculate the size of th ..."
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In this paper we find verifiable regularity conditions to ensure that Sestimators of scale and regression and MMestimators of regression are uniformly consistent and uniformly asymptotically normally distributed over contamination neighbourhoods. Moreover, we show how to calculate the size of these neighbourhoods. In particular, we find that, for MMestimators computed with Tukey’s family of bisquare score functions, there is a tradeoff between the size of these neighbourhoods and both the breakdown point of the Sestimators and the leverage of the contamination that is allowed in the neighbourhood. These results extend previous work of SalibianBarrera and Zamar for the locationscale model.
The OneWay Analysis Of Variance
, 2000
"... The oneway analysis of variance is concerned with comparisons of the locations of several onedimensional samples. This paper gives a simple unified approach to the problem based on location fits with associated bounds derived from a smooth location functional. The method is standardized for normal ..."
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The oneway analysis of variance is concerned with comparisons of the locations of several onedimensional samples. This paper gives a simple unified approach to the problem based on location fits with associated bounds derived from a smooth location functional. The method is standardized for normally distributed data with no assumptions placed on the sample sizes or the sample variances. The smoothness of the location functional leads to a stable analysis in that small perturbations either of the data or the standardizing model lead to only small changes in the analysis
DIFFERENTIABILITY OF MFUNCTIONALS OF LOCATION AND SCATTER BASED ON T LIKELIHOODS
"... Abstract. The paper aims at finding widely and smoothly defined nonparametric location and scatter functionals. As a convenient vehicle, maximum likelihood estimation of the location vector µ and scatter matrix Σ of an elliptically symmetric t distribution on R d with degrees of freedom ν > 1 ex ..."
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Abstract. The paper aims at finding widely and smoothly defined nonparametric location and scatter functionals. As a convenient vehicle, maximum likelihood estimation of the location vector µ and scatter matrix Σ of an elliptically symmetric t distribution on R d with degrees of freedom ν > 1 extends to an Mfunctional defined on all probability distributions P in a weakly open, weakly dense domain U. Here U consists of P putting not too much mass in hyperplanes of dimension < d, as shown for empirical measures by Kent and Tyler (Ann. Statist. 1991). It is shown here that (µ, Σ) is analytic on U, for the bounded Lipschitz norm, or for d = 1, for the sup norm on distribution functions. For k = 1, 2, ..., and other norms, depending on k and more directly adapted to t functionals, one has continuous differentiability of order k, allowing the deltamethod to be applied to (µ, Σ) for any P in U, which can be arbitrarily heavytailed. These results imply asymptotic normality of the corresponding Mestimators (µ n , Σ n ). In dimension d = 1 only, the t ν functional (µ, σ) extends to be defined and weakly continuous at all P .
Recall that a sequence Qk of laws (probability measures), here on R
, 2005
"... said to converge weakly toalaw Q if fdQk → fdQ for every bounded continuous function f. There exists a metric ρ on the set of all laws on R Q d which metrizes weak convergence, in other words Qk → Q weakly if and only if ρ(Qk,Q) → 0, e.g. Dudley (2002, Sec. 11.3). A set U of laws is called weakly o ..."
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said to converge weakly toalaw Q if fdQk → fdQ for every bounded continuous function f. There exists a metric ρ on the set of all laws on R Q d which metrizes weak convergence, in other words Qk → Q weakly if and only if ρ(Qk,Q) → 0, e.g. Dudley (2002, Sec. 11.3). A set U of laws is called weakly open if and only if whenever Q ∈ U and Qk → Q weakly we have k ∈ U for all k large enough. Equivalently, for each Q ∈ U, there is an r> 0 such that whenever ρ(Q, P) < r we have P ∈ U. Much of robustness theory emphasizes mixture laws P =(1 − λ)F0 + λQ (1) where Q is an arbitrary “contaminating ” distribution, F0 is a special distribution with a density, say for definiteness a normal, and 0 ≤ λ< 1/2, e.g. Huber [20, pp. 86, 89]. Despite the generality of Q, the contamination model (1) doesn’t include some, perhaps the majority, of laws P treated as
Explanation (science) vs. Prediction (engineering)
"... Statistics has no concept of approximation. Look up the word ‘approximation ’ in the index of any book on statistics. On the other hand, all models are approximate, useful,... Recognize this explicitly by removing all references to ‘truth’ for real data sets. Approximatio sine veritate – D. W. Müll ..."
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Statistics has no concept of approximation. Look up the word ‘approximation ’ in the index of any book on statistics. On the other hand, all models are approximate, useful,... Recognize this explicitly by removing all references to ‘truth’ for real data sets. Approximatio sine veritate – D. W. Müller (Kiefer–Müller process) The approximation of a data set xn = (x1,..., xn) by a model P which is a probability measure over Rn. 2Data and models Data xn = (x1,..., xn) ∈ Rn. Model P, a probability measure on Rn.
Uniform asymptotics for S and MMregression estimators Running title: “Uniform asymptotics for robust regression”
, 2008
"... In this paper we find verifiable regularity conditions to ensure that Sestimators of scale and regression and MMestimators of regression are uniformly consistent and uniformly asymptotically normally distributed over contamination neighbourhoods. Moreover, we show how to calculate the size of the ..."
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In this paper we find verifiable regularity conditions to ensure that Sestimators of scale and regression and MMestimators of regression are uniformly consistent and uniformly asymptotically normally distributed over contamination neighbourhoods. Moreover, we show how to calculate the size of these neighbourhoods. In particular, we find that, for MMestimators computed with Tukey’s family of bisquare score functions, there is a tradeoff between the size of these neighbourhoods and both the breakdown point of the Sestimators and the leverage of the contamination that is allowed in the neighbourhood. These results extend previous work of SalibianBarrera and Zamar for locationscale to the linear regression model. Key words: Robustness, robust inference, uniform asymptotics, robust regression. 1