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Estimation under l1-Symmetry

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  • Fourdrinier, Dominique
  • Lemaire, Anne-Sophie

Abstract

The estimation of the location parameter of an l1-symmetric distribution is considered. Specifically when a p-dimensional random vector has a distribution that is a mixture of uniform distributions on the l1-sphere, we investigate a general class of estimators of the form [delta]=X+g. Under the usual quadratic loss, domination of [delta] over X is obtained through the partial differential inequality 4 div g+2Xc[not partial differential]2g+ ||g||2[less-than-or-equals, slant]0 and a new superharmonicity-type-like notion adapted to the l1-context. Specifically the condition of l1-superharmonicity is that 2[Delta]f+Xc[backward difference]3f[less-than-or-equals, slant]0 and div [backward difference]3f[greater-or-equal, slanted]0 as compared to the usual (l2) condition [Delta]f[less-than-or-equals, slant]0.

Suggested Citation

  • Fourdrinier, Dominique & Lemaire, Anne-Sophie, 2002. "Estimation under l1-Symmetry," Journal of Multivariate Analysis, Elsevier, vol. 83(2), pages 303-323, November.
    • Handle: RePEc:eee:jmvana:v:83:y:2002:i:2:p:303-323
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    References listed on IDEAS

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    1. Ralescu, Stefan & Brandwein, Ann Cohen & Strawderman, William E., 1992. "Stein estimation for non-normal spherically symmetric location families in three dimensions," Journal of Multivariate Analysis, Elsevier, vol. 42(1), pages 35-50, July.
    2. Chou, Jine-Phone & Strawderman, William E., 1990. "Minimax estimation of means of multivariate normal mixtures," Journal of Multivariate Analysis, Elsevier, vol. 35(2), pages 141-150, November.
    3. Ann Brandwein & Stefan Ralescu & William Strawderman, 1993. "Shrinkage estimators of the location parameter for certain spherically symmetric distributions," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 45(3), pages 551-565, September.
    4. Kuwana, Yoichi & Kariya, Takeaki, 1991. "LBI tests for multivariate normality in exponential power distributions," Journal of Multivariate Analysis, Elsevier, vol. 39(1), pages 117-134, October.
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