Redescending $M$-Estimates of Multivariate Location and Scatter

Kent, John T.; Tyler, David E.

doi:10.1214/aos/1176348388

Cited by 179 publications

(167 citation statements)

References 8 publications

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“…The algorithm converges to a solution of (7) given any initial valueΣ 0 . The proof of convergence is analogous to the convergent proof for the non-regularized Mestimators given in [10] and is given in the Appendix. For convergence of the algorithm we need to assume that ρ(t) is continuously differentiable and satisfies Condition 1 (stated below in Section IV) and that the M -estimating equation (7) has a unique solution Σ.…”

Section: Regularized M -Estimators Of Scatter Matrixmentioning

confidence: 98%

Regularized <formula formulatype="inline"><tex Notation="TeX">$M$</tex> </formula>-Estimators of Scatter Matrix

Ollila

Tyler

2014

IEEE Trans. Signal Process.

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Abstract-In this paper, a general class of regularized Mestimators of scatter matrix are proposed which are suitable also for low or insufficient sample support (small n and large p) problems. The considered class constitutes a natural generalization of M -estimators of scatter matrix (Maronna, 1976) and are defined as a solution to a penalized M -estimation cost function that depend on a pair (α, β) of regularization parameters. We derive general conditions for uniqueness of the solution using concept of geodesic convexity. Since these conditions do not include Tyler's M -estimator, necessary and sufficient conditions for uniqueness of the penalized Tyler's cost function are established separately. For the regularized Tyler's M -estimator, we also derive a simple, closed form and data dependent solution for choosing the regularization parameter based on shape matrix matching in the mean squared sense. An iterative algorithm that converges to the solution of the regularized M -estimating equation is also provided. Finally, some simulations studies illustrate the improved accuracy of the proposed regularized M -estimators of scatter compared to their non-regularized counterparts in low sample support problems. An example of radar detection using normalized matched filter (NMF) illustrate that an adaptive NMF detector based on regularized M -estimators are able to maintain accurately the preset CFAR level and at at the same time provide similar probability of detection as the (theoretical) NMF detector.

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Section: Regularized M -Estimators Of Scatter Matrixmentioning

confidence: 98%

Regularized <formula formulatype="inline"><tex Notation="TeX">$M$</tex> </formula>-Estimators of Scatter Matrix

Ollila

Tyler

2014

IEEE Trans. Signal Process.

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“…So, applying Proposition 1 on the existence of Tyler's Mfunctional of scatter at P ⊗ U ν gives the following existence results for the t M-functionals. This proposition is given in Kent & Tyler (1991) for empirical distributions, and with only strict inequality in part (b). We state the general result here for completeness.…”

Section: The Multivariate T M-functionalsmentioning

confidence: 99%

“…Results for this case can be obtained from the scatter-only problem by using an identity introduced in Kent & Tyler (1991). This identity relates the t M-functionals of location and scatter in q dimensions with parameter ν to a scatter only t M-functional in q+1 dimension with parameter ν −1.…”

Section: The Multivariate T M-functionalsmentioning

confidence: 99%

“…That is, it is proportional to Tyler's M-functional at P ⊗ δ 1 . The identity as presented in Kent & Tyler (1991) is for empirical distributions only. However, it is straightforward to note that the identity applies to any arbritrary P .…”

Section: The Multivariate T M-functionalsmentioning

confidence: 99%

“…Outside of the t M-estimates, the relationship between a simultaneous M-estimate of location and scatter and a "monotonic" Mestimate of scatter only does not generally hold, nor does the uniqueness of the simultaneous Mestimates of location and scatter generally hold, see Kent & Tyler (1991). Studying the breakdown behavior of the simultaneous M-estimates location and scatter in general may require studying them as minimizers of objective functions, which is the approach used in Kent & Tyler (1991) with the right-hand side going to Q(x + V ) as R → ∞. This implies Lemma 12 holds.…”

Section: Non-integer Values Of ν and More General M-functionalsmentioning

confidence: 99%

See 2 more Smart Citations

On the Breakdown Properties of Some Multivariate M-Functionals*

Dümbgen

Tyler

2005

Scand J Stat

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ABSTRACT. For probability distributions on IR q , a detailed study of the breakdown properties of some multivariate M-functionals related to Tyler's (1987a) "distribution-free" M-functional of scatter is given. These include a symmetrized version of Tyler's M-functional of scatter, and the multivariate t M-functionals of location and scatter. It is shown that for "smooth" distributions, the (contamination) breakdown point of Tyler's M-functional of scatter and of its symmetrized version are 1/q and 1 − 1 − 1/q, respectively. For the multivariate t M-functional which arises from the maximum likelihood estimate for the parameters of an elliptical t distribution on ν ≥ 1 degrees of freedom the breakdown point at smooth distributions is 1/(q + ν). Breakdown points are also obtained for general distributions, including empirical distributions. Finally, the sources of breakdown are investigated. It turns out that breakdown can only be caused by contaminating distributions that are concentrated near low-dimensional subspaces.

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Robust Estimation of Multivariate Location and Scatter

Maronna¹,

Yohai²

2005

Encyclopedia of Statistical Sciences

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Redescending $M$-Estimates of Multivariate Location and Scatter

Cited by 179 publications

References 8 publications

Regularized <formula formulatype="inline"><tex Notation="TeX">$M$</tex> </formula>-Estimators of Scatter Matrix

Regularized <formula formulatype="inline"><tex Notation="TeX">$M$</tex> </formula>-Estimators of Scatter Matrix

On the Breakdown Properties of Some Multivariate M-Functionals*

Robust Estimation of Multivariate Location and Scatter

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