Gogtas, Hakan
(2004)
Improving coverage of rectangular confidence interval.
Doctoral Dissertation, University of Pittsburgh.
(Unpublished)
Abstract
To find a better confidence region is always of interest in statistics. One way to find better confidence regions is to uniformly improve coverage probability over the usual confidence region while maintaining the same volume. Thus, the classical spherical confidence regions for the mean vector of a multivariate normal distribution have been improved by changing the point estimator for the parameterIn 1961, James and Stein found a shrinkage estimator having total mean square error, TMSE, smaller than that of the usual estimator. In 1982, Casella and Hwang gave an analytical proof of the dominance of the confidence sphere which uses the James Stein estimator as its center over the usual confidence sphere centered at the sample mean vector. This opened up new possibilities in multiple comparisonsThis dissertation will focus on simultaneous confidence intervals for treatment means and for the differences between treatment means and the mean of a control in oneway and twoway Analysis of Variance, ANOVA, studies. We make use of Steintype shrinkage estimators as centers to improve the simultaneous coverage of those confidence intervals. The main obstacle to an analytic study is that the rectangular confidence regions are not rotation invariant like the spherical confidence regions Therefore, we primarily use simulation to show dominance of the rectangular confidence intervals centered around a shrinkage estimator over the usual rectangular confidence regions centered about the sample means. For the oneway ANOVA model, our simulation results indicate that our confidence procedure has higher coverage probability than the usual confidence procedure if the number of means is sufficiently large. We develop a lower bound for the coverage probability of our rectangular confidence region which is a decreasing function of the shrinkage constant for the estimator used as center and use this bound to prove that the rectangular confidence intervals centered around a shrinkage estimator have coverage probability uniformly exceeding that of the usual rectangular confidence regions up to an arbitrarily small epsilon when the number of means is sufficiently large. We show that these intervals have strictly greater coverage probability when all the parameters are zero, and that the coverage probability of the two procedures converge to one another when at least one of the parameters becomes arbitrarily largeTo check the reliability of our simulations for the oneway ANOVA model, we use numerical integration to calculate the coverage probability for the rectangular confidence regions. Gaussian quadrature making use of Hermite polynomials is used to approximate the coverage probability of our rectangular confidence regions for n=2, 3, 4. The difference in results between numerical integration and simulations is negligible. However, numerical integration yields values slightly higher than the simulationsA similar approach is applied to develop improved simultaneous confidence intervals for the comparison of treatment means with the mean of a control. We again develop a lower bound for the coverage probability of our confidence procedure and prove results similar to those that we proved for the oneway ANOVA model. We also apply our approach to develop improved simultaneous confidence intervals for the cell means for a twoway ANOVA model. We again primarily use simulation to show dominance of the rectangular confidence intervals centered around an appropriate shrinkage estimator over the usual rectangular confidence regions. We again develop a lower bound for the coverage probabilities of our confidence procedure and prove the same results that we proved for the oneway model
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Details
Item Type: 
University of Pittsburgh ETD

Status: 
Unpublished 
Creators/Authors: 

ETD Committee: 

Date: 
23 September 2004 
Date Type: 
Completion 
Defense Date: 
6 July 2004 
Approval Date: 
23 September 2004 
Submission Date: 
16 August 2004 
Access Restriction: 
No restriction; Release the ETD for access worldwide immediately. 
Institution: 
University of Pittsburgh 
Schools and Programs: 
Dietrich School of Arts and Sciences > Statistics 
Degree: 
PhD  Doctor of Philosophy 
Thesis Type: 
Doctoral Dissertation 
Refereed: 
Yes 
Uncontrolled Keywords: 
numerical integration; oneway anova model; shrinkage estimator; Simultaneous confidence intervals; twoway anova model 
Other ID: 
http://etd.library.pitt.edu/ETD/available/etd08162004211317/, etd08162004211317 
Date Deposited: 
10 Nov 2011 19:59 
Last Modified: 
15 Nov 2016 13:49 
URI: 
http://dscholarship.pitt.edu/id/eprint/9118 
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