An admissible estimator in the one-parameter exponential family with ambiguous information

Research paper by Yasushi Nagata

Indexed on: 01 Dec '83Published on: 01 Dec '83Published in: Annals of the Institute of Statistical Mathematics


LetX be a random variable from the one-parameter exponential family with the probability element β(θ) exp (θx)dm(x) for which an ambiguous prior information is available to the effect that θ is likely to be larger than or equal to a known constant. The information is represented by a fuzzy set with the membership function χ(θ). Then it is shown that\({{X + \int_{ - \infty }^\infty {\chi '(\theta )\beta (\theta )\exp (\theta X)d\theta } } \mathord{\left/ {\vphantom {{X + \int_{ - \infty }^\infty {\chi '(\theta )\beta (\theta )\exp (\theta X)d\theta } } {\int_{ - \infty }^\infty {\chi '(\theta )\beta (\theta )\exp (\theta X)d\theta } }}} \right. \kern-\nulldelimiterspace} {\int_{ - \infty }^\infty {\chi '(\theta )\beta (\theta )\exp (\theta X)d\theta } }}\) is an admissible estimator for E0(X) under the quadratic loss function.