Validity of the parametric bootstrap for goodness-of-fit testing in semiparametric models

Genest, Christian; Rémillard, Bruno

Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 6, pp. 1096-1127.

Pour tester qu’une loi $P$ donnée provient d’une famille paramétrique $𝒫$, on est souvent amené à comparer une estimation non paramétrique $A_n$ d’une fonctionnelle $A$ de $P$ à un élément $A_{{\theta }_n}$ correspondant à une estimation ${\theta }_n$ de $\theta$. Dans bien des cas, la loi asymptotique de statistiques de tests bâties à partir du processus $n^{1/2}(A_n-A_{{\theta }_n})$ dépend de la loi inconnue $P$. On montre ici que si les suites $A_n$ et ${\theta }_n$ 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 $A_n^{*}$ et ${\theta }_n^{*}$ sont des analogues de $A_n$ et ${\theta }_n$ déduits d’un échantillon de loi $P_{{\theta }_n}$, les processus empiriques $n^{1/2}(A_n-A_{{\theta }_n})$ et $n^{1/2}(A_n^{*}-A_{{\theta }_n^{*}})$ 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 $P$ belongs to a parameterized family $𝒫$, one is often led to compare a nonparametric estimate $A_n$ of some functional $A$ of $P$ with an element $A_{{\theta }_n}$ corresponding to an estimate ${\theta }_n$ of $\theta$. In many cases, the asymptotic distribution of goodness-of-fit statistics derived from the process $n^{1/2}(A_n-A_{{\theta }_n})$ depends on the unknown distribution $P$. It is shown here that if the sequences $A_n$ and ${\theta }_n$ of estimators are regular in some sense, a parametric bootstrap approach yields valid approximations for the $P$-values of the tests. In other words if $A_n^{*}$ and ${\theta }_n^{*}$ are analogs of $A_n$ and ${\theta }_n$ computed from a sample from $P_{{\theta }_n}$, the empirical processes $n^{1/2}(A_n-A_{{\theta }_n})$ and $n^{1/2}(A_n^{*}-A_{{\theta }_n^{*}})$ 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.

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