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January, 1976 Robust $M$-Estimators of Multivariate Location and Scatter
Ricardo Antonio Maronna
Ann. Statist. 4(1): 51-67 (January, 1976). DOI: 10.1214/aos/1176343347


Let $\mathbf{x}_1,\cdots, \mathbf{x}_n$ be a sample from an $m$-variate distribution which is spherically symmetric up to an affine transformation. This paper deals with the robust estimation of the location vector $\mathbf{t}$ and scatter matrix $\mathbf{V}$ by means of "$M$-estimators," defined as solutions of the system: $\sum_i u_1(d_i)(\mathbf{x}_i - \mathbf{t}) = \mathbf{0}$ and $n^{-1}\sum_i u_2(d_i^2)(\mathbf{x}_i - \mathbf{t})(\mathbf{x}_i - \mathbf{t})' = \mathbf{V}$, where $d_i^2 = (\mathbf{x}_i - \mathbf{t})'\mathbf{V}^{-1}(\mathbf{x}_i - \mathbf{t})$. Existence and uniqueness of solutions of this system are proved under general assumptions about the functions $u_1$ and $u_2$. Then the estimators are shown to be consistent and asymptotically normal. The breakdown bound and the influence function are calculated, showing some weaknesses of the estimates for high dimensionality. An algorithm for the numerical calculation of the estimators is described. Finally, numerical values of asymptotic variances, and Monte Carlo small-sample results are exhibited.


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Ricardo Antonio Maronna. "Robust $M$-Estimators of Multivariate Location and Scatter." Ann. Statist. 4 (1) 51 - 67, January, 1976.


Published: January, 1976
First available in Project Euclid: 12 April 2007

zbMATH: 0322.62054
MathSciNet: MR388656
Digital Object Identifier: 10.1214/aos/1176343347

Keywords: $M$-estimators , G2G35 , G2H99 , outlier detection , robust estimation , scatter matrix

Rights: Copyright © 1976 Institute of Mathematical Statistics


Vol.4 • No. 1 • January, 1976
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