A Mixture Approach to Bayesian Goodness of Fit
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(3). CREST, INSEE, and Universite´ Paris 5, 75232 Paris cedex 05 Summary. We consider a Bayesian approach to goodness of fit, that is, to the problem of testing whether or not a given parametric model is compatible with the data at hand. We thus conwhere denotes a cumulative distribution function sider a parametric family with parameter . The null hypothesis is for an unknown , that is, there exists such that . If does not hold, is a random variable on which is not distributed as . The alternative nonparametric hypothesis can thus be interpreted being distributed from a general cdf on , where is infinite dimensional. as Instead of using a functional basis as in Verdinelli and Wasserman (1998), we represent as the (infinite) mixture of Beta distributions, Estimation within both parametric and nonparametric structures are implemented using MCMC algorithms that estimate the number of components in the mixture. Since we are concerned with a goodness of fit problem, it is more of interest to consider a functional distance to the tested model as the basis of our test, rather than the corresponding Bayes factor, since the later puts more emphasis on the parameters. We therefore propose a new test procedure based on , with both an asymptotic justification and a finite sampler implementation.. !#"%$ &')(+*-,. /0*#12,435/768 :91 &)' /0*#1 /768 :91 35/768 ;91 /0*#1 <%= /768 ;91 > <%= ? '@35/768 :91BAC/D9FEG?':1 HJIKL? IMON /0P I D Q I 1SR. T 7/ 2 D1 U%V W T /YX D1;Z *\[^]. AMS 1991 classification. Primary 62C05. Secondary 60J05, 62F15, 65D30, 65C60.. _4a^` b:cd a` e. ´ ´ ` ´ Nous considerons une nouvelle approche bayesienne des problemes d’adequation ` parametrique ´ ´ de loi, a` savoir la compatibilite´ d’un modele avec un echantillon. Si la famille ´ ee ´ s’ecrit ´ ´ ´ ee ´ par parametr ou` est une fonction de repartition parametr ` , l’hypothese nulle est avec inconnu. Donc il existe tel que ´ . Si n’est pas vrai, est une variable aleatoire sur qui n’est pas dis´ comme ´ ´ tribuee quelque soit . L’alternative non-parametrique peut donc s’interpreter ´ suivant une loi gen ´ erale ´ comme distribuee sur , ou` est de dimension infinie. Au lieu de faire appel a` une base fonctionnelle comme dans Verdinelli et Wasser´ ´ ´ man (1998), nous representons comme un melange (infini) de lois beta ` ´ ´ L’estimation des modeles parametrique et non-parametrique se fait ´ ´ via des algorithmes MCMC qui evaluent le nombre de composantes dans le melange. Pour ´ ´ ´ ´ evaluer l’adequation a` la loi, il nous semble plus interessant de considerer une distance fonc` teste, ´ ˆ que le facteur de Bayes, dependant ´ tionnelle au modele , plutot plus directement ` ´ ´ sur des parametres. Nous proposons une nouvelle procedure de test fondee , ´ en fournissant une justification asymptotique et une mise en œuvre a` taille dechantillon finie.. fgh i jk"%$ &)'l(\*m,- )& ' /0*#1 35/768 :91 /0*#1 <n= ?':1 HJIKL? IMON /0P I DQ I 1SR 53 /768 :91. T /72 D1. <n=. /768 ;91. /768 :91 >. /0*#1\,. ?'@35/768 :91A/D9oE. U%V W T /YX D1;Z *\[8]. ´ Inference ´ ´ ´ ´ Mots-cles: bayesienne, melanges de lois beta, processus de vie et mort, consis´ ` a` dimension variable tence, algorithmes MCMC, estimation non-parametrique, modele Keywords: Bayesian inference, Beta mixture distribution, birth-and-death process, consistency, MCMC algorithms, nonparametric estimation, variable dimension model. piqsrt uwvSxt@y z x{ { |j}@~ v vrt@y : |j FG y2rt us o 8 G+F^2: 66 hwr oS8Y0 7¡¢ ¡£¤ Y0 ¦¥ 0 7§ ¢ ¨©i¦ª^:£¥O«;¬¦ª ©®F¡¢ ¯¥ ¦¥%D«n¤^0 7¢°%Y± ².
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