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Computing Bayesian predictive distributions: The K square and K prime distributions

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Niveau: Supérieur, Doctorat, Bac+8
ha l-0 01 13 70 5, v er sio n 1 - 1 4 N ov 2 00 6 Computing Bayesian predictive distributions: The K-square and K-prime distributions 14th November 2006 Jacques Poitevineau1, Bruno Lecoutre2 1ERIS and LAM/LCPE, UMR 7604, C.N.R.S., Universite Paris 6 et Ministere de la Culture, 11 rue de Lourmel, 75015 Paris, France. Email: 2ERIS, and Laboratoire de Mathematiques Raphael Salem, UMR 6085, C.N.R.S. et Universite de Rouen, Avenue de l'Universite, BP 12, 76801 Saint-Etienne-du-Rouvray, France. E-mail: Abstract The computation of two Bayesian predictive distributions which are discrete mixtures of incomplete beta functions is considered. The num- ber of iterations can easily become large for these distributions and thus, the accuracy of the result can be questionable. Therefore, existing algo- rithms for that class of mixtures are improved by introducing round-off error calculation into the stopping rule. A further simple modification is proposed to deal with possible underflows that may prevent recurrence to work properly. Keywords: Predictive distribution; Bayesian approach; Round-off error; Incomplete beta function 1 Introduction The K-square and K-prime distributions have been introduced in Lecoutre (1984).

  • square distribu- tion

  • bayesian approach

  • gamma functions

  • computing bayesian predictive

  • prime distribution

  • free- dom parameters

  • alternate chi-square


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ComputingBayesianpredictivedistributions:TheK-squareandK-primedistributions14thNovember2006JacquesPoitevineau1,BrunoLecoutre21ERISandLAM/LCPE,UMR7604,C.N.R.S.,Universite´Paris6etMiniste`redelaCulture,11ruedeLourmel,75015Paris,France.Email:poitevin@ccr.jussieu.fr2ERIS,andLaboratoiredeMathe´matiquesRaphae¨lSalem,UMR6085,C.N.R.S.etUniversite´deRouen,Avenuedel’Universite´,BP12,76801Saint-Etienne-du-Rouvray,France.E-mail:bruno.lecoutre@univ-rouen.frAbstractThecomputationoftwoBayesianpredictivedistributionswhicharediscretemixturesofincompletebetafunctionsisconsidered.Thenum-berofiterationscaneasilybecomelargeforthesedistributionsandthus,theaccuracyoftheresultcanbequestionable.Therefore,existingalgo-rithmsforthatclassofmixturesareimprovedbyintroducinground-offerrorcalculationintothestoppingrule.Afurthersimplemodificationisproposedtodealwithpossibleunderflowsthatmaypreventrecurrencetoworkproperly.Keywords:Predictivedistribution;Bayesianapproach;Round-offerror;Incompletebetafunction1IntroductionTheK-squareandK-primedistributionshavebeenintroducedinLecoutre(1984).TheycanbecharacterizedasmixturesoftheclassicalnoncentralFandnoncentraltdistributionsrespectively(Lecoutre,1999).ThesetwodistributionsareinvolvedintheBayesianpredictiveapproachforplanningandmonitoringexperiments(Lecoutre,2001).Inparticular,theyareusefultoolsforsamplesizedetermination,usingthepredictivedistributionsoftheteststatisticsandofthelimitsofconfidenceintervalsunderstandardnormalmodels,assuminga1