Bayesian tolerance intervals for variance component models
Abstract
English: 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 Jeffreys’ 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. Afrikaans: In enige vervaardigings proses, het die verbetering van gehalte essensieel geword.
Aangesien die waargenome variasie in ’n proses ’n funksie van die gehalte van die
vervaardigde items is, bied die beraming van variansie komponente en toleransie intervalle,
’n metode waardeur die waargenome proses variasie geëvalueer kan word.
In teenstelling met vertrouens intervalle wat slegs inligting rakende ’n onbekende populasie
parameter bied, bied toleransie intervalle inligting rakende die populasie in sy
geheel. Die statistiese probleem met betrekking tot die afleiding van gevolgtrekkings
uit kwantiele van waarskynlikheids verdelings wat veronderstel is om ’n proses genoegsaam
te beskryf, word dus deur toleransie intervalle aangespreek. LuidensWolfinger
(1998), is die ( ; ) toleransie interval (waar die inhoud en die vertroue van die
interval is), die - verwagtings toleransie interval (waar die verwagte oordekking
van die interval is) en die vooraf vasgestelde toleransie interval (waar die interval
reeds vasgestel is en die persentasie proses waarnemings wat hierin voorkom, beraam
word), die drie toleransie intervalle wat meestal gebruik word. In die geval van die
gebalanseerde een rigting toevallige effekte model, het Wolfinger (1998) ’n simulasie
gebaseerde beskouing vir die bepaling van Bayesiaanse toleransie intervalle voorgestel.
In hierdie proefskrif, word Wolfinger (1998) se voorgestelde Bayesiaanse simulasie
metode vir die bepaling van die drie algemene toleransie intervalle, toegepas vir die
beraming van toleransie intervalle in die gevalle van die gebalanseerde enkelveranderlike
normaal model, die gebalanseerde een rigting toevallige effekte model met
N(0; 2
" ) verdeelde foute, die gebalanseerde een rigting toevallige effekte model met
student t - verdeelde foute en die gebalanseerde geneste toevallige effekte model. Die voorgestelde modelle sal toegepas word op data stelle afkomstig uit verskillende
terreine. Dit sluit data stelle in aangaande gelykheids mates gemeet op keramiek
parte, die hoeveelheid aktiewe bestandeel teenwoordig in klein gegroepeerde stelle
medisinale tablette, die hoeveelheid yster konsentraat in deeltjies per miljoen teenwoordig,
bepaal deur emissie spektroskopie, en ’n eg Suid - Afrikaanse data stel aangaande
die persentasie toename in lengte van ’n aaneenlopende poliëster vesel
voordat dit breek. Die Suid - Afrikaanse data stel is deur Prof. Nico Laubscher by
SANS Fibres (Pty.) Ltd. versamel. Daarbenewens word metodes vir die vergelyking
van twee of meer kwantiele, in die geval van die gebalanseerde enkelveranderlike
normaal model, voorgestel. Bykomend, word Wolfinger (1998) se simulasie metode
aangepas om die beraming van toleransie intervalle in die geval van die gemiddeld
van waarnemings uit nuwe of onbekende gegroepeerde stelle in te sluit. Deur van die
Bayesiaanse simulasie metode gebruik te maak vir die voorgestelde toevallige effekte
model met student t - verdeelde foute, word die identifisering van moontlike uitskieters
ook geïllustreer. Die gebruik van spesifieke prior inligting is een van die voordele van
die voorgestelde Bayesiaanse simulasie metode. Dit is egter juis die gebruik van hierdie
prior inligting wat wyd veroordeel word, aangesien algemene nie - inligtende prior
verdelings, soos ’n Jeffreys’ prior, ’n dramatiese onverwagte uitwerking op die posterior
verdeling tot gevolg kan hê as meer as een parameter ter sprake is. Ter erkenning
van die probleem, word daar gewys dat die nie - inligtende prior verdelings, voorgestel
vir die kwantiele en inhoud van die vooraf vasgestelde toleransie intervalle in die
gevalle van die enkelveranderlike normaal model, die voorgestelde toevallige effekte
model vir die gemiddeld van waarnemings uit onbekende of nuwe gegroepeerde
stelle en die gebalanseerde geneste twee rigting toevallige effekte model, beide verwysings
priors (soos voorgestel deur Berger en Bernardo (1992c)) en waarskynlikheids
ooreenstemmende priors (soos voorgestel deur Datta en Ghosh (1995)), is. Aangesien
al die voorgestelde modelle goed gevaar het vir die bepaling van toleransie intervalle,
is die unieke en buigsame kenmerke van die Bayesiaanse simulasie metode
geïllustreer.
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