Compact hypothesis and extremal set estimators

Mexia, João ; Corte Real, Pedro

Discussiones Mathematicae. Probability and Statistics, Tome 23 (2003) no. 2, pp. 103-121

Résumé

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 β⃗₀⃗^k is the sole point in ∇, strongly consistent pointwise estimators, β̂⃗̂ₙ̂⃗̂^k: n ∈ ℕ for β⃗₀⃗^k are derived and confidence ellipsoids for β⃗₀⃗^k centered at β̂⃗̂ₙ̂⃗̂^k are obtained, as well as, strongly consistent tests. Lastly an application to binary data is presented.

Détail

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