Sankhya: The Indian Journal of Statistics

1997, Volume 59, Series A, Pt. 3, 366-375

ON $\Gamma$-MINIMAX ESTIMATION WITH ORDERED OBSERVATIONS

By

BRANI VIDAKOVIC, Duke University, Durham

SUMMARY. In this note we derive $\Gamma$-minimax estimators for the location parameter of the normal and uniform models. All distributions with uniformly bounded variances and means belonging to a fixed interval form a class of priors, $\Gamma$. We restrict ourselves to the rules which are linear combinations of order statistics. Optimal decision rules in such decision-theoretic framework are, as expected, rules linear in the minimal sufficient statistics, $\bar X$ and $(X_{1:n}, X_{n:n}$). However in the ``no intercept''- class of decision rules, the optimal $\Gamma$-minimax rule requires knowledge of all order statistics.

 AMS (1991) subject classification. 62C20, 62G30

Key words and phrases. $\Gamma$-minimax rule, order statistics, rules linear in the order statistics

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