$M$-functionals of multivariate scatter

Duembgen, Lutz; Pauly, Markus; Schweizer, T.

doi:10.1214/15-ss109

Cited by 22 publications

(31 citation statements)

References 41 publications

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“…Section 3 presents some analytical properties of the latter functional which are essential to understand existing algorithms and to devise new ones. These parts follow closely a recent survey of multivariate M -functionals by Dümbgen et al [5]. In Section 4 we discuss the aforementioned fixed-point algorithm of Kent and Tyler [8] and explain rigorously why it is suboptimal.…”

Section: Introductionmentioning

confidence: 82%

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New algorithms for M-estimation of multivariate scatter and location

Dümbgen

Nordhausen

Schuhmacher

2016

Journal of Multivariate Analysis

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We present new algorithms for M -estimators of multivariate scatter and location and for symmetrized M -estimators of multivariate scatter. The new algorithms are considerably faster than currently used fixed-point and other algorithms. The main idea is to utilize a Taylor expansion of second order of the target functional and devise a partial Newton-Raphson procedure. In connection with symmetrized M -estimators we work with incomplete U -statistics to accelerate our procedures initially. *

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Section: Introductionmentioning

confidence: 82%

“…The assumptions on ρ and Q imply that the functional L(·, Q) has essentially a unique minimizer (see [8], [2] or [5]): Theorem 1. In Setting 0 there exists a unique matrix Σ 0 (Q) ∈ R q×q sym,>0 such that L 0 (Σ 0 (Q), Q) ≤ L 0 (·, Q) and det(Σ 0 (Q)) = 1.…”

Section: The General Settingsmentioning

confidence: 99%

New algorithms for M-estimation of multivariate scatter and location

Dümbgen

Nordhausen

Schuhmacher

2016

Journal of Multivariate Analysis

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“…For more details on M estimators we refer to Dümbgen, Pauly, and Schweizer (2015) and Dümbgen, Nordhausen, and Schuhmacher (2016).…”

Section: Plug-in Estimator For Standard Modelmentioning

confidence: 99%

pvclass: An R Package for p Values for Classification

Zumbrunnen¹,

Dümbgen²

2017

J. Stat. Soft.

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Let (X, Y ) be a random variable consisting of an observed feature vector X and an unobserved class label Y ∈ {1, 2, . . . , L} with unknown joint distribution. In addition, let D be a training data set consisting of n completely observed independent copies of (X, Y ). Instead of providing point predictors (classifiers) for Y , we compute for each b ∈ {1, 2, . . . , L} a p value π b (X, D) for the null hypothesis that Y = b, treating Y temporarily as a fixed parameter, i.e., we construct a prediction region for Y with a certain confidence. The advantages of this approach over more traditional ones are reviewed briefly. In principle, any reasonable classifier can be modified to yield nonparametric p values.We describe the R package pvclass which computes nonparametric p values for the potential class memberships of new observations as well as cross-validated p values for the training data. Additionally, it provides graphical displays and quantitative analyses of the p values.

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“…Dümbgen's M-functional is also a TR-version of symmetrized spatial sign matrix. The very recent paper (Dümbgen et al, 2015) considers a framework to generalize M-functionals based on symmmetrizations of arbitrary order. Our TR Gini covariance matrix can be treated as an example of their Case 1 with the symmetrization of order 2.…”

Section: Theorem 22mentioning

confidence: 99%

“…Dümbgen (1998) considered symmetrized TR spatial sign covariance matrix on the difference of two independent vectors X 1 − X 2 . Dümbgen et al (2015) provided a general treatment on M-functionals of scatter based on symmetrizations of arbitrary order. Our TR Gini covariance matrix turns out to be a pairwise symmetrized scatter M-functional.…”

Section: Introductionmentioning

confidence: 99%

Gini covariance matrix and its affine equivariant version

Dang

Sang

Weatherall³

2016

Stat Papers

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We propose a new covariance matrix called Gini covariance matrix (GCM), which is a natural generalization of univariate Gini mean difference (GMD) to the multivariate case. The extension is based on the covariance representation of GMD by applying the multivariate spatial rank function. We study properties of GCM, especially in the elliptical distribution family. In order to gain the affine equivariance property for GCM, we utilize the transformation-retransformation (TR) technique and obtain an affine equivariant version GCM that turns out to be a symmetrized M-functional. The influence function of those two GCM's are obtained and their estimation has been presented. Asymptotic results of estimators have been established. A closely related scatter Kotz functional and its estimator are also explored. Finally, asymptotical efficiency and finite sample efficiency of the TR version GCM are compared with those of sample covariance matrix, Tyler-M estimator and other scatter estimators under different distributions.

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$M$-functionals of multivariate scatter

Cited by 22 publications

References 41 publications

New algorithms for M-estimation of multivariate scatter and location

New algorithms for M-estimation of multivariate scatter and location

pvclass: An R Package for p Values for Classification

Gini covariance matrix and its affine equivariant version

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