## BAYESIAN TOLERANCE INTERVALS FOR VARIANCE COMPONENT MODELS

The improvement of quality has become a very important part of any manufacturing
process. Since variation observed in a process is a function of the quality of the manufactured
items, estimating variance components and tolerance intervals present a
method for evaluating process variation. As apposed to confidence intervals that provide
information concerning an unknown population parameter, tolerance intervals
provide information on the entire population, and, therefore address the statistical
problem of inference about quantiles and other contents of a probability distribution
that is assumed to adequately describe a process. According to Wolfinger (1998),
the three kinds of commonly used tolerance intervals are, the (; ) tolerance interval
(where is the content and is the confidence), the - expectation tolerance
interval (where is the expected coverage of the interval) and the fixed - in - advance
tolerance interval in which the interval is held fixed and the proportion of process
measurements it contains, is estimated. Wolfinger (1998) presented a simulation
based approach for determining Bayesian tolerance intervals in the case of the balanced
one - way random effects model. In this thesis, the Bayesian simulation method
for determining the three kinds of tolerance intervals as proposed by Wolfinger (1998)
is applied for the estimation of tolerance intervals in a balanced univariate normal
model, a balanced one - way random effects model with standard N(0; 2
" ) measurement
errors, a balanced one - way random effects model with student t - distributed
measurement errors and a balanced two - factor nested random effects model. The
proposed models will be applied to data sets from a variety of fields including flatness measurements measured on ceramic parts, measuring the amount of active ingredient
found in medicinal tablets manufactured in small batches, measurements of iron
concentration in parts per million determined by emission spectroscopy and a South
- African data set collected at SANS Fibres (Pty.) Ltd. concerned with measuring the
percentage increase in length before breaking of continuous filament polyester. In
addition, methods are proposed for comparing two or more quantiles in the case
of the balanced univariate normal model. Also, the Bayesian simulation method proposed
by Wolfinger (1998) for the balanced one - way random effects model will be
extended to include the estimation of tolerance intervals for averages of observations
from new or unknown batches. The Bayesian simulation method proposed for determining
tolerance intervals for the balanced one - way random effects model with student
t - distributed measurement errors will also be used for the detection of possible
outlying part measurements. One of the main advantages of the proposed Bayesian
approach, is that it allows explicit use of prior information. The use of prior information
for a Bayesian analysis is however widely criticized, since common non - informative
prior distributions such as a JeffreysÃ¢ prior can have an unexpected dramatic effect on
the posterior distribution. In recognition of this problem, it will also be shown that the
proposed non - informative prior distributions for the quantiles and content of fixed
- in - advance tolerance intervals in the cases of the univariate normal model, the
proposed random effects model for averages of observations from new or unknown
batches and the balanced two - factor nested random effects model, are reference
priors (as proposed by Berger and Bernardo (1992c)) as well as probability matching
priors (as proposed by Datta and Ghosh (1995)). The unique and flexible features of
the Bayesian simulation method were illustrated since all mentioned models performed
well for the determination of tolerance intervals.

 Identifer oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:ufs/oai:etd.uovs.ac.za:etd-08152012-115551 Date 15 August 2012 Creators Hugo, Johan Contributors Prof AJ van der Merwe Publisher University of the Free State Source Sets South African National ETD Portal Language en-uk Detected Language English Type text Format application/pdf Source http://etd.uovs.ac.za//theses/available/etd-08152012-115551/restricted/ Rights unrestricted, I hereby certify that, if appropriate, I have obtained and attached hereto a written permission statement from the owner(s) of each third party copyrighted matter to be included in my thesis, dissertation, or project report, allowing distribution as specified below. I certify that the version I submitted is the same as that approved by my advisory committee. I hereby grant to University Free State or its agents the non-exclusive license to archive and make accessible, under the conditions specified below, my thesis, dissertation, or project report in whole or in part in all forms of media, now or hereafter known. I retain all other ownership rights to the copyright of the thesis, dissertation or project report. I also retain the right to use in future works (such as articles or books) all or part of this thesis, dissertation, or project report.

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