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Abstract:
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From a sequence of observations, the ones that exceed previous ones in a time seriesare called records. The pioneer paper of record theory is considered to be Chandler [39]. Thistheory gained its popularity doe to significant public interest towards records. As a result, largenumber of papers are published on this topic.Record values are very important in statistics. Record values are applied in parameterestimation issues, characterization issues, hypothesis and stationarity tests, etc. Also, theirusefulness in probability theory and in theory of random process is tremendous.This dissertation discusses applications of records through numerical evaluations of maximumlikelihood estimators of parameters of the three-parameter extensions of Weibull distributionfamily, new recurrence relations of record moments, records in Bayesian inference, applicationsof records in characterization issues for random chord length distributions as well with theasymptotic behaviour of extremes of random chord lengths. This dissertation consists on sixchapters.Several examples of records are presented in the first chapter.Second chapter discusses the strict formulations of records from a sequence of independentand identically distributed random variables. Their application and their extensions from thesame model are presented, as well with several interesting results.The problem of existence and uniqueness of maximum likelihood estimators based on recordsis elaborated in the third chapter. In this chapter, we present sufficient conditions that confirmthe existence and uniqueness of maximum likelihood estimators for a three-parameter extensionsof Weibull distributions. Also, several well known results are presented as examples. Severalresults from this chapter could be found in [135].The fourth chapter is dedicated to moment recurrence relations of a three-parameter ex-tension of Weibull distribution based on records with possible applications. These results arepublished in [136].Fifth chapter deals with Bayesian prediction of order statistics based on record values. Here,we expand the applicability of records in real problems and provide a better understanding oftheir significance. Several results presented in this chapter could be found in [136].In the sixth chapter the random chord length issue is considered through the record valuetheory. A new generation method of random chords is presented. The study of limit behaviourof maximum length of random chords for all cases of generation is also conducted. Character-ization results for random chord length distributions based on record moments are obtained.
Several results presented in this chapter could be found in [134]. |