302 research outputs found
Spatial Sign Correlation
A new robust correlation estimator based on the spatial sign covariance
matrix (SSCM) is proposed. We derive its asymptotic distribution and influence
function at elliptical distributions. Finite sample and robustness properties
are studied and compared to other robust correlation estimators by means of
numerical simulations.Comment: 20 pages, 7 figures, 2 table
A robust partial least squares method with applications
Partial least squares regression (PLS) is a linear regression technique developed to relate many
regressors to one or several response variables. Robust methods are introduced to reduce or
remove the effect of outlying data points. In this paper we show that if the sample covariance
matrix is properly robustified further robustification of the linear regression steps of the PLS
algorithm becomes unnecessary. The robust estimate of the covariance matrix is computed by
searching for outliers in univariate projections of the data on a combination of random directions
(Stahel-Donoho) and specific directions obtained by maximizing and minimizing the kurtosis
coefficient of the projected data, as proposed by Peña and Prieto (2006). It is shown that this
procedure is fast to apply and provides better results than other procedures proposed in the
literature. Its performance is illustrated by Monte Carlo and by an example, where the algorithm is
able to show features of the data which were undetected by previous methods
Depth weighted scatter estimators
General depth weighted scatter estimators are introduced and investigated.
For general depth functions, we find out that these affine equivariant scatter
estimators are Fisher consistent and unbiased for a wide range of multivariate
distributions, and show that the sample scatter estimators are strong and
\sqrtn-consistent and asymptotically normal, and the influence functions of the
estimators exist and are bounded in general. We then concentrate on a specific
case of the general depth weighted scatter estimators, the projection depth
weighted scatter estimators, which include as a special case the well-known
Stahel-Donoho scatter estimator whose limiting distribution has long been open
until this paper. Large sample behavior, including consistency and asymptotic
normality, and efficiency and finite sample behavior, including breakdown point
and relative efficiency of the sample projection depth weighted scatter
estimators, are thoroughly investigated. The influence function and the maximum
bias of the projection depth weighted scatter estimators are derived and
examined. Unlike typical high-breakdown competitors, the projection depth
weighted scatter estimators can integrate high breakdown point and high
efficiency while enjoying a bounded-influence function and a moderate maximum
bias curve. Comparisons with leading estimators on asymptotic relative
efficiency and gross error sensitivity reveal that the projection depth
weighted scatter estimators behave very well overall and, consequently,
represent very favorable choices of affine equivariant multivariate scatter
estimators.Comment: Published at http://dx.doi.org/10.1214/009053604000000922 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
An Object-Oriented Framework for Robust Multivariate Analysis
Taking advantage of the S4 class system of the programming environment R, which facilitates the creation and maintenance of reusable and modular components, an object-oriented framework for robust multivariate analysis was developed. The framework resides in the packages robustbase and rrcov and includes an almost complete set of algorithms for computing robust multivariate location and scatter, various robust methods for principal component analysis as well as robust linear and quadratic discriminant analysis. The design of these methods follows common patterns which we call statistical design patterns in analogy to the design patterns widely used in software engineering. The application of the framework to data analysis as well as possible extensions by the development of new methods is demonstrated on examples which themselves are part of the package rrcov.
A robust partial least squares method with applications
Partial least squares regression (PLS) is a linear regression technique developed to relate many regressors to one or several response variables. Robust methods are introduced to reduce or remove the effect of outlying data points. In this paper we show that if the sample covariance matrix is properly robustified further robustification of the linear regression steps of the PLS algorithm becomes unnecessary. The robust estimate of the covariance matrix is computed by searching for outliers in univariate projections of the data on a combination of random directions (Stahel-Donoho) and specific directions obtained by maximizing and minimizing the kurtosis coefficient of the projected data, as proposed by Peña and Prieto (2006). It is shown that this procedure is fast to apply and provides better results than other procedures proposed in the literature. Its performance is illustrated by Monte Carlo and by an example, where the algorithm is able to show features of the data which were undetected by previous methods.
Robust Estimators are Hard to Compute
In modern statistics, the robust estimation of parameters of a regression hyperplane is a central problem. Robustness means that the estimation is not or only slightly affected by outliers in the data. In this paper, it is shown that the following robust estimators are hard to compute: LMS, LQS, LTS, LTA, MCD, MVE, Constrained M estimator, Projection Depth (PD) and Stahel-Donoho. In addition, a data set is presented such that the ltsReg-procedure of R has probability less than 0.0001 of finding a correct answer. Furthermore, it is described, how to design new robust estimators. --Computational statistics,complexity theory,robust statistics,algorithms,search heuristics
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