Compact Hypothesis and Extremal Set Estimators

n extremal estimation theory the estimators are local or absolute extremes of functions defined on the cartesianproduct of the parameter by the sample space. Assuming that these functions converge uniformly, in a convenientstochastic way, to a limit function g, set estimators for the set r of absolute maxima (minima) of g are obtained underthe compactness assumption that r is contained in a known compact U. A strongly consistent test is presented forthis assumption. Moreover, when the true parameter value ~ k0 is the sole point in r, strongly consistent pointwiseestimators, {ˆ~ kn : n 2 N} for ~ k0 are derived and confidence ellipsoids for ~ k0 centered atˆ~ kn are obtained, as well as,strongly consistent tests. Lastly an application to binary data is presented.Keywords: Extremal estimators, set estimators, confidence ellipsoids, strong