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Diagnostic testing and evaluation of maximum likelihood models. (English) Zbl 0591.62094
Cet article traite d’une methode générale de test de spécification de modèles statistiques. Le cadre est le suivant: \(Y_ 1,..,Y_ n\) sont les observations, i.i.d. Y, v.a., dépendant d’un paramètre \(\theta\). G est la fonction de répartition vraie de Y tandis que \(F(y,\theta)\) est la famille de distributions adoptée pour estimer \(\theta\). \({\hat\theta}_ n\) est l’estimateur du maximum de vraisemblance associé.
Les tests de spécification étudiés ici sont fondés sur des statistiques du type: \({\hat \tau}_ n=n^{-1}\sum^{n}_{1}c(Y_ i,{\hat \theta}_ n)\), les c(y,\(\theta)\) étant des fonctions critères vérifiant (\(\forall \theta):\int c(y,\theta)\quad dF(y,\theta)=0.\)
L’intérêt de la statistique \({\hat\tau}_ n\) est que si le modèle est correctement spécifié, il existe \(\theta_ 0\) tel que \(F'(y,\theta_ 0)\) soit une version de la densité vraie g(y) et par suite: \({\hat \tau}_ n\to^{p.s.}E(c(Y_ i,\theta_ 0))=0.\) Au contraire, en cas d’erreur de spécification, il existe \({\bar\tau}\) (en général \(\neq 0)\) tel que \({\hat \tau}_ n\to^{p.s.}{\bar\tau}.\)
Sous certaines hypothèses est établie la distribution asymptotique du couple (\({\hat \theta}{}_ n,{\hat \tau}_ n)\) ainsi que le comportement local de la limite p.s. \({\bar \tau}\) de \({\hat \tau}{}_ n\) en cas d’erreur de spécification dans une direction v(y) (càd si \(dG(y)=(1+v(y))dF(y,\theta_ 0))\). Un exemple numèrique est fourni, montrant comment, au prix d’un simple test T, on peut pratiquement tester la spécification d’un modèle économétrique.
Reviewer: V.Cohen

MSC:
62P20 Applications of statistics to economics
62F03 Parametric hypothesis testing
62E20 Asymptotic distribution theory in statistics
62H15 Hypothesis testing in multivariate analysis
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