Minimaxschätzer für den mittelwert ϑ einer normalverteilten zufallsgröβe mit bekannter varianz bei vorgegebener oberer und unterer schranke fiir ϑ

Abstract

Let X ₁,…,X _n be independently distributed with the same normal distribution with known variance σ¹. The problem is to given a minimax point estimator for the unknown mean ϑ. Let the los function s be defined by s(ϑ, d) = (ϑ-d)² , and let the domain Ω of ϑ and the set D of possible decisions be the finite interval [a, b]. a<b. The problem of finding minimax estimators with respect to the general point estimation problem is reduced to the special problem charecterized by n = 1, σ¹ = 1, Ω = [0,c], c>0. Minimax point estimators are determined if c∊ (0, 1.2]. For further values of c (minimax estimators are evaluated numerically. We finally compare certain estimators (minimax, maximum likelihood, etc.) including a graphical representation of their risk-functions.

Keywords:

• Estimation problem
• normal distribution
• quadratic loss function
• minimax estimator
• least favorable distribution