Discussiones Mathematicae Probability and Statistics 23(2) (2003) 103-121

COMPACT HYPOTHESIS AND EXTREMAL SET ESTIMATORS

João Tiago Mexia and Pedro Corte Real

Universidade Nova de Lisboa, Departamento de Matemática
da Faculdade de Ciências e Tecnologia
Quinta da Torre, 2825-114 Monte de Caparica, Portugal
e-mail:
parcr@mail.fct.unl.pt

Abstract

In extremal estimation theory the estimators are local or absolute extremes of functions defined on the cartesian product of the parameter by the sample space. Assuming that these functions converge uniformly, in a convenient stochastic way, to a limit function g, set estimators for the set ∇ of absolute maxima (minima) of g are obtained under the compactness assumption that ∇ is contained in a known compact U. A strongly consistent test is presented for this assumption. Moreover, when the true parameter value [(β )\vec]0k is the sole point in ∇ , strongly consistent pointwise estimators, { [^([(β )\vec])]nk: n ∈ N } for [(β )\vec]0k are derived and confidence ellipsoids for [(β )\vec]0k centered at [^([(β )\vec])]nk are obtained, as well as, strongly consistent tests. Lastly an application to binary data is presented.

Keywords: extremal estimators, set estimators, confidence ellipsoids, strong consistency, binary data.

2000 Mathematics Subject Classification: 60F15, 62F10.

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Received 24 November 2002
Revised 3 November 2003