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

Hooper, Peter Michael

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Permalink

https://hdl.handle.net/2142/68341

Description

Title

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

Author(s)

Hooper, Peter Michael

Issue Date

1981

Department of Study

Statistics

Discipline

Statistics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

Statistics

Language

eng

Abstract

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.

Wijsman (1980, Multivariate Analysis V) describes a method for generating smallest simultaneous confidence sets {A(,i)} for a family of parametric functions {(psi)(,i)((gamma))} starting from a confidence set C(,0) for (gamma). In addition the method determines the confidence set C(,1) (R-HOOK EQ) C(,0) with respect to which the family {A(,i)} is exact; i.e., {(gamma) (epsilon) C(,1)(X)} = {(psi)i((gamma)) (epsilon) Ai(X) (FOR ALL)i}. Confidence sets C(,1) satisfying the above equation for some family {A(,i)} are termed self-reproducing.(,)

The thesis extends Wijsman's method to include randomized set estimators and applies the method to find all self-reproducing set estimators defined in terms of an invariantly sufficient function, together with the corresponding smallest simultaneous set estimators. The applications are given in the general multivariate analysis of variance (GMANOVA) model of Potthoff and Roy under full group reduction and in the MANOVA model under both full and partial group reduction. The parameter (gamma) is a q x p matrix M and the families of parametric functions considered are {a'M, a (epsilon) R('q)}, {Mb, b (epsilon) R('p)}, {a'Mb}, and {tr N'M, N:q x p}. As an example, the confidence set determined by the extension of Roy's maximum root criterion to GMANOVA was found to be self-reproducing for all the above families.

The confidence set determined by the step-down procedure in MANOVA is not self-reproducing. Thus its confidence coefficent gives a conservative lower bound on the probability of simultaneous coverage by the induced family of simultaneous confidence sets. The thesis develops an improved approximation for this probability.

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