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- W1998524376 abstract "In the context of Huber's theory of robust estimation of a location parameter, the literature on minimax properties of $M$-, $R$- and $L$-estimators is surveyed. New results are obtained for the model in which the unknown error distribution is assumed to lie in a Levy neighbourhood of a symmetric distribution $G: mathscr{P}_{varepsilon,delta}(G) = {Fmid G(x - delta) - varepsilon leq F(x) leq G(x + delta) + varepsilon text{for all} x}$. Under reasonably general conditions on $G$, the distribution $F_0$ in $mathscr{P}_{varepsilon,delta}(G)$ which minimizes Fisher information for location is found. Huber's minimax property for $M$-estimators is shown to hold for $R$-estimators but to fail for $L$-estimators in Levy neighbourhoods. The latter is proved by constructing a subneighbourhood of distributions $mathscr{F}_0$, with $F_0 in mathscr{F}_0 subset mathscr{P}_{varepsilondelta}(G)$, such that the asymptotic variance of the $L$-estimator which is asymptotically efficient at $F_0$ is minimized over $mathscr{F}_0$ at $F_0$." @default.
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- W1998524376 date "1989-03-01" @default.
- W1998524376 modified "2023-09-26" @default.
- W1998524376 title "Minimax Properties of $M-, R-$ and $L$-Estimators of Location in Levy Neighbourhoods" @default.
- W1998524376 doi "https://doi.org/10.1214/aos/1176347020" @default.
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