dc.contributor.author Modarres-Mousavi, Shabnam en dc.date.accessioned 2014-03-14T20:07:35Z en dc.date.available 2014-03-14T20:07:35Z en dc.date.issued 2006-01-17 en dc.identifier.other etd-02182006-145606 en dc.identifier.uri http://hdl.handle.net/10919/26235 en dc.description.abstract Our objective is to monitor the changes in a proportion with correlated binary observations. All of the published work on this subject used the first-order Markov chain model for the data. Increasing the order of dependence above one by extending a standard Markov chain model entails an exponential increase of both the number of parameters and the dimension of the transition probability matrix. In this dissertation, we develop a particular Markov chain structure, the Multilevel Model (MLM), to model the correlation between binary data. The basic idea is to assign a lower probability to observing a 1 when all previous correlated observations are 0's, and a higher probability to observing a 1 as the last observed 1 gets closer to the current observation. We refer to each of the distinct situations of observing a 1 as a "level". For a given order of dependence, , at most different values of conditional probabilities of observing a 1 can be assigned. So the number of levels is always less than or equal to . Compared to a direct extension of the first-order Markov model to higher orders, our model is considerably parsimonious. The number of parameters for the MLM is only one plus the number of levels, and the transition probability matrix is . We construct a CUSUM control chart for monitoring a proportion with correlated binary observations. First, we use the probability structure of a first-order Markov chain to derive a log-likelihood ratio based CUSUM control statistic. Then, we model this CUSUM statistic itself as a Markov chain, which in turn allows for designing a control chart with specified statistical properties: the Markov Binary CUSUM (MBCUSUM) chart. We generalize the MBCUSUM to account for any order of dependence between binary observations through implying MLM to the data and to our CUSUM control statistic. We verify that the MBCUSUM has a better performance than a curtailed Shewhart chart. Also, we show that except for extremely large changes in the proportion (of interest) the MBCUSUM control chart detects the changes faster than the Bernoulli CUSUM control chart, which is designed for independent observations. en dc.publisher Virginia Tech en dc.relation.haspart STAT_Dissertation.pdf en dc.rights In Copyright en dc.rights.uri http://rightsstatements.org/vocab/InC/1.0/ en dc.subject Markov chain models en dc.subject CUSUM control chart en dc.subject Correlated observations en dc.subject Binary data. en dc.subject Monitoring a proportion en dc.title Monitoring Markov Dependent Binary Observations with a Log-Likelihood Ratio Based CUSUM Control Chart en dc.type Dissertation en dc.contributor.department Statistics en dc.description.degree Ph. D. en thesis.degree.name Ph. D. en thesis.degree.level doctoral en thesis.degree.grantor Virginia Polytechnic Institute and State University en thesis.degree.discipline Statistics en dc.contributor.committeechair Reynolds, Marion R. Jr. en dc.contributor.committeemember Vining, G. Geoffrey en dc.contributor.committeemember Schulman, Robert S. en dc.contributor.committeemember Woodall, William H. en dc.contributor.committeemember Birch, Jeffrey B. en dc.identifier.sourceurl http://scholar.lib.vt.edu/theses/available/etd-02182006-145606/ en dc.date.sdate 2006-02-18 en dc.date.rdate 2008-04-04 en dc.date.adate 2006-04-04 en
﻿