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An asymptotic analysis, when the sample size n tends to infinity, of the optimal confidence region established in Czarnowska and Nagaev (2001) is considered. As a result, two confidence regions, both close to the optimal one when n is sufficiently large, are suggested with a mild assumption on the distribution of a location-scale family.
The maximum likelihood scale invariant estimator of the shape parameter of the gamma distribution, proposed by the authors [Statist. Probab. Lett. 78 (2008)], is considered. The asymptotics of the mean square error of this estimator, with respect to that of the usual maximum likelihood estimator, is established.
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