Pour tester qu’une loi donnée provient d’une famille paramétrique , on est souvent amené à comparer une estimation non paramétrique d’une fonctionnelle de à un élément correspondant à une estimation de . Dans bien des cas, la loi asymptotique de statistiques de tests bâties à partir du processus dépend de la loi inconnue . On montre ici que si les suites et d’estimateurs sont régulières dans un sens précis, le recours au rééchantillonnage paramétrique conduit à des approximations valides des seuils des tests. Autrement dit si et sont des analogues de et déduits d’un échantillon de loi , les processus empiriques et convergent alors conjointement en loi vers des copies indépendantes de la même limite. Ce résultat est employé pour valider l’approche par rééchantillonnage paramétrique dans le cadre de tests d’adéquation pour des familles de lois et de copules multivariées. Deux types de tests sont envisagés : les uns comparent la version empirique d’une loi ou d’une copule et son estimation paramétrique sous l’hypothèse nulle ; les autres mesurent la distance entre les estimations paramétrique et non paramétrique de la loi associée à la transformation intégrale de probabilité classique. La validité du rééchantillonnage à deux degrés est aussi démontrée dans les cas où l’estimation paramétrique est difficile à calculer. La méthodologie est illustrée au moyen d’un nouveau test d’adéquation de copules fondé sur une fonctionnelle de Cramér-von Mises du processus de copule empirique.
In testing that a given distribution belongs to a parameterized family , one is often led to compare a nonparametric estimate of some functional of with an element corresponding to an estimate of . In many cases, the asymptotic distribution of goodness-of-fit statistics derived from the process depends on the unknown distribution . It is shown here that if the sequences and of estimators are regular in some sense, a parametric bootstrap approach yields valid approximations for the -values of the tests. In other words if and are analogs of and computed from a sample from , the empirical processes and then converge jointly in distribution to independent copies of the same limit. This result is used to establish the validity of the parametric bootstrap method when testing the goodness-of-fit of families of multivariate distributions and copulas. Two types of tests are considered: certain procedures compare the empirical version of a distribution function or copula and its parametric estimation under the null hypothesis; others measure the distance between a parametric and a nonparametric estimation of the distribution associated with the classical probability integral transform. The validity of a two-level bootstrap is also proved in cases where the parametric estimate cannot be computed easily. The methodology is illustrated using a new goodness-of-fit test statistic for copulas based on a Cramér-von Mises functional of the empirical copula process.
Mots clés : copula, goodness-of-fit test, Monte Carlo simulation, parametric bootstrap, P-values, semiparametric estimation
@article{AIHPB_2008__44_6_1096_0, author = {Genest, Christian and R\'emillard, Bruno}, title = {Validity of the parametric bootstrap for goodness-of-fit testing in semiparametric models}, journal = {Annales de l'I.H.P. Probabilit\'es et statistiques}, pages = {1096--1127}, publisher = {Gauthier-Villars}, volume = {44}, number = {6}, year = {2008}, doi = {10.1214/07-AIHP148}, mrnumber = {2469337}, zbl = {1206.62044}, language = {en}, url = {http://www.numdam.org/articles/10.1214/07-AIHP148/} }
TY - JOUR AU - Genest, Christian AU - Rémillard, Bruno TI - Validity of the parametric bootstrap for goodness-of-fit testing in semiparametric models JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2008 SP - 1096 EP - 1127 VL - 44 IS - 6 PB - Gauthier-Villars UR - http://www.numdam.org/articles/10.1214/07-AIHP148/ DO - 10.1214/07-AIHP148 LA - en ID - AIHPB_2008__44_6_1096_0 ER -
%0 Journal Article %A Genest, Christian %A Rémillard, Bruno %T Validity of the parametric bootstrap for goodness-of-fit testing in semiparametric models %J Annales de l'I.H.P. Probabilités et statistiques %D 2008 %P 1096-1127 %V 44 %N 6 %I Gauthier-Villars %U http://www.numdam.org/articles/10.1214/07-AIHP148/ %R 10.1214/07-AIHP148 %G en %F AIHPB_2008__44_6_1096_0
Genest, Christian; Rémillard, Bruno. Validity of the parametric bootstrap for goodness-of-fit testing in semiparametric models. Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 6, pp. 1096-1127. doi : 10.1214/07-AIHP148. http://www.numdam.org/articles/10.1214/07-AIHP148/
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