Smallest Simultaneous Confidence Sets, Using Sufficiency and Invariance, With Applications to Manova and Gmanova

The thesis first describes how sufficiency and invariance considerations can be applied in problems of confidence set estimation to reduce the class of set estimators under investigation. Let X be a random variables taking values in with distribution P(,(theta)), (theta) (epsilon) (THETA), and suppose a confidence set is desired for (gamma) = (gamma)((theta)), where (gamma) takes values in (GAMMA). The main tool used is the representation of a randomized set estimator as a function (phi) : x (GAMMA) (--->) {0, 1}. Sufficiency is defined in terms of the family {P(,(theta),(gamma)):((theta), (gamma)) (epsilon) (THETA) x (GAMMA)} of distributions on x (GAMMA), where P(,(theta),(gamma)) is P(,(theta)) supported on x {(gamma)}. Let T: x (GAMMA) (--->) and let ('T) be the family of induced distributions on . Let S be a function defined on which is sufficient for ('T). Then the class of randomized set estimators based on S is essentially complete among those based on T provided the risk function depends only on E(,(theta))(phi)(X,(gamma)), ((theta),(gamma)) (epsilon) (THETA) x (GAMMA). In applications of interest the above definition of sufficiency is equivalent to the usual one given in terms of {P(,(theta)):(theta) (epsilon) (THETA)}. If G is an invariance group acting on the problem and T above is a maximal invariant under G, then the principle of invariance allows one to restrict attention to set estimators based on the invariantly sufficient function S. Moreover, if G acts transitively on (THETA), then S is an invariant pivotal quantity.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Statistics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2014

Author dc:creator

• Hooper, Peter Michael

• Statistics

Language dc:language

eng

Identifier

(UMI)AAI8114435

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/68341