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22
Breakdown and Groups
, 2002
"... The concept of breakdown point was... In this paper we argue that this success is intimately connected to the fact that the translation and affine groups act on the sample space and give rise to a definition of equivariance for statistical functionals. For such functionals a nontrivial upper bound f ..."
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Cited by 24 (6 self)
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The concept of breakdown point was... In this paper we argue that this success is intimately connected to the fact that the translation and affine groups act on the sample space and give rise to a definition of equivariance for statistical functionals. For such functionals a nontrivial upper bound for the breakdown point can be shown. In the absence of such a group structure a breakdown point of one is attainable and this is perhaps the decisive reason why the concept of breakdown point in other situations has not proved as successful. Even if a natural group is present it is often not sufficiently large to allow a nontrivial upper bound for the breakdown point. One exception to this is the problem of the autocorrelation structure of time series where we derive a nontrivial upper breakdown point using the group of realizable linear filters. The paper is formulated in an abstract manner to emphasize the role of the group and the resulting equivariance structure
Predictive spatiotemporal models for spatially sparse environmental data. Statist. Sinica 15 547–568. MR2190219
 Ann. Statist
, 2005
"... Abstract: We present a family of spatiotemporal models which are geared to provide timeforward predictions in environmental applications where data is spatially sparse but temporally rich. That is measurements are made at few spatial locations (stations), but at many regular time intervals. When p ..."
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Cited by 16 (8 self)
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Abstract: We present a family of spatiotemporal models which are geared to provide timeforward predictions in environmental applications where data is spatially sparse but temporally rich. That is measurements are made at few spatial locations (stations), but at many regular time intervals. When predictions in the time direction is the purpose of the analysis, then spatialstationarity assumptions which are commonly used in spatial modeling, are not necessary. The family of models proposed does not make such assumptions and consists of a vector autoregressive (VAR) specification, where there are as many time series as stations. However, by taking into account the spatial dependence structure, a model building strategy is introduced which borrows its simplicity from the BoxJenkins strategy for univariate autoregressive (AR) models for time series. As for AR models, model building may be performed either by displaying sample partial correlation functions, or by minimizing an information criterion. A simulation study illustrates the gain resulting from our modeling strategy. Two environmental data sets are studied. In particular, we find evidence that a parametric modeling of the spatiotemporal correlation function is not appropriate because it rests on too strong assumptions. Moreover, we propose to compare model selection strategies with an outofsample validation method based on recursive prediction errors. Key words and phrases: Accumulated prediction errors, spatiotemporal correlation, partial correlation, vector autoregression. 1.
Robust estimation of the scale and of the autocovariance function of gaussian short and longrange dependent processes
 Journal of Time Series Analysis
, 2011
"... Abstract. A desirable property of an autocovariance estimator is to be robust to the presence of additive outliers. It is wellknown that the sample autocovariance, being based on moments, does not have this property. Hence, the use of an autocovariance estimator which is robust to additive outliers ..."
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Cited by 6 (2 self)
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Abstract. A desirable property of an autocovariance estimator is to be robust to the presence of additive outliers. It is wellknown that the sample autocovariance, being based on moments, does not have this property. Hence, the use of an autocovariance estimator which is robust to additive outliers can be very useful for timeseries modeling. In this paper, the asymptotic properties of the robust scale and autocovariance estimators proposed by Rousseeuw and Croux (1993) and Ma and Genton (2000) are established for Gaussian processes, with either shortrange or longrange dependence. It is shown in the shortrange dependence setting that this robust estimator is asymptotically normal at the rate √ n, where n is the number of observations. An explicit expression of the asymptotic variance is also given and compared to the asymptotic variance of the classical autocovariance estimator. In the longrange dependence setting, the limiting distribution displays the same behavior than that of the classical autocovariance estimator, with a Gaussian limit and rate √ n when the Hurst parameter H is less 3/4 and with a nonGaussian limit (belonging to the second Wiener chaos) with rate depending on the Hurst parameter when H ∈ (3/4, 1). Some MonteCarlo experiments are presented to illustrate our claims and the Nile River data is analyzed as an application. The theoretical results and the empirical evidence strongly suggest using the robust estimators as an alternative to estimate the dependence structure of Gaussian processes. 1.
Highly Robust Estimation of Dispersion Matrices
, 2001
"... In this paper, we propose a new componentwise estimator of a dispersion matrix, based on a highly robust estimator of scale. The key idea is the elimination of a location estimator in the dispersion estimation procedure. The robustness properties are studied by means of the influence function and th ..."
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Cited by 5 (0 self)
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In this paper, we propose a new componentwise estimator of a dispersion matrix, based on a highly robust estimator of scale. The key idea is the elimination of a location estimator in the dispersion estimation procedure. The robustness properties are studied by means of the influence function and the breakdown point. Further characteristics such as asymptotic variance and efficiency are also analyzed. It is shown in the componentwise approach, for multivariate Gaussian distributions, that covariance matrix estimation is more difficult than correlation matrix estimation. The reason is that the asymptotic variance of the covariance estimator increases with increasing dependence, whereas it decreases with increasing dependence for correlation estimators. We also prove that the asymptotic variance of dispersion estimators for multivariate Gaussian distributions is proportional to the asymptotic variance of the underlying scale estimator. The proportionality value depends only on the underlying dependence. Therefore, the highly robust dispersion estimator is among the best robust choice at the present time in the componentwise approach, because it is locationfree and combines small variability and robustness properties such as high breakdown point and bounded influence function. A simulation study is carried out in order to assess the behavior of the new estimator. First, a comparison with another robust componentwise estimator based on the median absolute deviation scale estimator is performed. The highly robust properties of the new estimator are confirmed. A second comparison with global estimators such as the method of moment estimator, the minimum volume ellipsoid, and the minimum covariance determinant estimator is also performed, with two types of outliers. In this case, the highly robust dispersion matrix estimator turns out to be an interesting compromise between the high efficiency of the method of moment estimator in noncontaminated situations and the highly robust properties of the minimum volume ellipsoid and minimum covariance determinant estimators in contaminated
Robustness properties of dispersion estimators
, 1999
"... In this paper, we derive the in uence function of dispersion estimators, based on a scale approach. The relation between the grosserror sensitivity of dispersion estimators and the one of the underlying scale estimator is described. We show that for the bivariate Gaussian distributions, the asympto ..."
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Cited by 5 (2 self)
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In this paper, we derive the in uence function of dispersion estimators, based on a scale approach. The relation between the grosserror sensitivity of dispersion estimators and the one of the underlying scale estimator is described. We show that for the bivariate Gaussian distributions, the asymptotic variance of covariance estimators is minimal in the independent case, and is strictly increasing with the absolute value of the underlying covariance. The behavior of the asymptotic variance of correlation estimators seems to be the opposite, i.e. maximal for independent data, and strictly decreasing with the absolute value of the underlying correlation. In particular, dispersion estimators based on Mestimators of scale are studied closely. The one based on the median absolute deviation is the most Brobust in the class of symmetric estimators. Some other examples are analyzed, based on the maximum likelihood and the Welsch estimator of scale.
1Robust Trend Estimation for AR(1) Disturbances
"... Abstract: We discuss the robust estimation of a linear trend if the noise follows an autoregressive process of first order. We find the ordinary repeated median to perform well except for negative correlations. In this case it can be improved by a PraisWinsten transformation using a robust autocorr ..."
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Cited by 3 (3 self)
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Abstract: We discuss the robust estimation of a linear trend if the noise follows an autoregressive process of first order. We find the ordinary repeated median to perform well except for negative correlations. In this case it can be improved by a PraisWinsten transformation using a robust autocorrelation estimator. Zusammenfassung: Wir behandeln die robuste Schätzung eines linearen Trends bei autoregressiven Fehlern erster Ordnung. Die Repeated Median
ESTIMATING SPECTRAL DENSITY FUNCTIONS ROBUSTLY
"... • We consider in the following the problem of robust spectral density estimation. Unfortunately, conventional spectral density estimators are not robust in the presence of additive outliers (cf. [18]). In order to get a robust estimate of the spectral density function, it turned out that cleaning t ..."
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Cited by 1 (0 self)
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• We consider in the following the problem of robust spectral density estimation. Unfortunately, conventional spectral density estimators are not robust in the presence of additive outliers (cf. [18]). In order to get a robust estimate of the spectral density function, it turned out that cleaning the time series in a robust way first and calculating the spectral density function afterwards leads to encouraging results. To meet these needs of cleaning the data we use a robust version of the Kalman filter which was proposed by Ruckdeschel ([26]). Similar ideas were proposed by Martin and Thomson ([18]). Both methods were implemented in R (cf. [23]) and compared by extensive simulation experiments. The competitive method is also applied to real data. As a special practical application we focus on actual heart rate variability measurements of diabetes patients. KeyWords:
Central limit theorem for the robust logregression wavelet estimation of the memory parameter in the gaussian semiparametric context,” Preprint, [Available Online
, 2011
"... Abstract. In this paper, we study robust estimators of the memory parameter d of a (possibly) non stationary Gaussian time series with generalized spectral density f. This generalized spectral density is characterized by the memory parameter d and by a function f ∗ which specifies the shortrange d ..."
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Abstract. In this paper, we study robust estimators of the memory parameter d of a (possibly) non stationary Gaussian time series with generalized spectral density f. This generalized spectral density is characterized by the memory parameter d and by a function f ∗ which specifies the shortrange dependence structure of the process. Our setting is semiparametric since both f ∗ and d are unknown and d is the only parameter of interest. The memory parameter d is estimated by regressing the logarithm of the estimated variance of the wavelet coefficients at different scales. The two estimators of d that we consider are based on robust estimators of the variance of the wavelet coefficients, namely the square of the scale estimator proposed by [27] and the median of the square of the wavelet coefficients. We establish a Central Limit Theorem for these robust estimators as well as for the estimator of d based on the classical estimator of the variance proposed by [19]. Some MonteCarlo experiments are presented to illustrate our claims and compare the performance of the different estimators. The properties of the three estimators are also compared on the Nile River data and the Internet traffic packet counts data. The theoretical results and the empirical evidence strongly suggest using the robust estimators as an alternative to estimate the memory parameter d of Gaussian time series. 1.
Comprehensive Definitions of BreakdownPoints for Independent and Dependent Observations
, 2000
"... We provide a new definition of breakdown in finite samples with an extension to asymptotic breakdown. Previous definitions center around defining a critical region for either the parameter or the objective function. If for a particular outlier constellation the critical region is entered, breakdown ..."
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We provide a new definition of breakdown in finite samples with an extension to asymptotic breakdown. Previous definitions center around defining a critical region for either the parameter or the objective function. If for a particular outlier constellation the critical region is entered, breakdown is said to occur. In contrast to the traditional approach, we leave the definition of the critical region implicit. Our definition encompasses all previous definitions of breakdown in both linear and nonlinear regression settings. In some cases, it leads to a different notion of breakdown than other procedures available. An advantage is that our new definition also applies to models for dependent observations (timeseries, spatial statistics) where current breakdown definitions typically fail. We illustrate our points using examples from linear and nonlinear regression as well as timeseries and spatial statistics.