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by: Providenci Mosciski Sr.


Providenci Mosciski Sr.
GPA 3.66

Peter Guttorp

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Peter Guttorp
Class Notes
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This 2 page Class Notes was uploaded by Providenci Mosciski Sr. on Wednesday September 9, 2015. The Class Notes belongs to STAT 583 at University of Washington taught by Peter Guttorp in Fall. Since its upload, it has received 23 views. For similar materials see /class/192506/stat-583-university-of-washington in Statistics at University of Washington.




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Date Created: 09/09/15
STAT 583 SPRING 2008 Robustness De nitions A sequence T n of estimators is qualitatively robust at F F 0 if the sequence of mappings taking F into the distribution of Tn under F is equicontinuous with respect to a suitable metric The in uence curve ICx le F 5x F measures the in uence of an additional observation x on the estimator TFn given a large sample from F The breakdown point of T is the smallest fraction of gross errors that carries the estimator beyond all bounds The gross error sensitivity GES of T is sup l ICx l An estimator is B robust if it has bounded GES The ef ciency of T at the distribution F is 11FJICx2 dF x where 11F is the Fisher information in a single observation from F An M estimator is the solution to Iwx6an x 0 A location M estimator has wx6 wx 6 Theorem The breakdown point of a location Mestimator is 1 1n woo v o w oo woo 39 wheren maX Let sx3990be the score function for a nominal distribution E E Fen De ne 7x sxeo a Where y hx lt y hm hx y s hx s z z hx gt z and a is chosen to make 7 integrate to zero with respect to dFo Theorem Among all location Mestimators with deFo 0 IWxsx60 dF0x 0 and Z su S c that based on 7 minimizes wzdF ls dF It is up to I lwxsxeodFox i w a multiplicative constant the unique such estimator the most ef cient location M estimator with this bound on its GES Theorem The most B robust estimator ie having the smallest GES is the median Regression Let The in uence curve for least squares regression is with known scalel ICT xy y XTTZSCXT71X unbounded both in residuals and in in uential points An Mestimator of regression for known scale is given by Inca y xTTFdFx y 0 Its in uence function is ICx y 77x y xTTFM 1n M where Mm F jn oc y xTTF xdeFx y a and n x r nx r Br The Mallows family of robust regression Mestimators has 71x r wxlr The breakdown point for regression Mestimators cannot exceed lp where pdimx A regression estimator with 50 breakdown is Siegel s repeated median estimator yr y x whose slope is given for bivariate regression by 3 medl med and intercept by medl yi 319 This can be generalized to higher dimensions but can be computationally somewhat tedious


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