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

Christian Genest; Bruno Rémillard

Annales de l'I.H.P. Probabilités et statistiques (2008)

  • Volume: 44, Issue: 6, page 1096-1127
  • ISSN: 0246-0203

Abstract

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In testing that a given distribution Pbelongs to a parameterized family 𝒫 , one is often led to compare a nonparametric estimateAn of some functional A of P with an element Aθn corresponding to an estimate θn of θ. In many cases, the asymptotic distribution of goodness-of-fit statistics derived from the process n1/2(An−Aθn) depends on the unknown distribution P. It is shown here that if the sequences An and θn 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 An* and θn* are analogs of An and θn computed from a sample from Pθn, the empirical processes n1/2(An−Aθn) and n1/2(An*−Aθn*) 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.

How to cite

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Genest, Christian, and Rémillard, Bruno. "Validity of the parametric bootstrap for goodness-of-fit testing in semiparametric models." Annales de l'I.H.P. Probabilités et statistiques 44.6 (2008): 1096-1127. <http://eudml.org/doc/78005>.

@article{Genest2008,
abstract = {In testing that a given distribution Pbelongs to a parameterized family $\mathcal \{P\}$, one is often led to compare a nonparametric estimateAn of some functional A of P with an element Aθn corresponding to an estimate θn of θ. In many cases, the asymptotic distribution of goodness-of-fit statistics derived from the process n1/2(An−Aθn) depends on the unknown distribution P. It is shown here that if the sequences An and θn 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 An* and θn* are analogs of An and θn computed from a sample from Pθn, the empirical processes n1/2(An−Aθn) and n1/2(An*−Aθn*) 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.},
author = {Genest, Christian, Rémillard, Bruno},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {copula; goodness-of-fit test; Monte Carlo simulation; parametric bootstrap; P-values; semiparametric estimation; -values},
language = {eng},
number = {6},
pages = {1096-1127},
publisher = {Gauthier-Villars},
title = {Validity of the parametric bootstrap for goodness-of-fit testing in semiparametric models},
url = {http://eudml.org/doc/78005},
volume = {44},
year = {2008},
}

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
PB - Gauthier-Villars
VL - 44
IS - 6
SP - 1096
EP - 1127
AB - In testing that a given distribution Pbelongs to a parameterized family $\mathcal {P}$, one is often led to compare a nonparametric estimateAn of some functional A of P with an element Aθn corresponding to an estimate θn of θ. In many cases, the asymptotic distribution of goodness-of-fit statistics derived from the process n1/2(An−Aθn) depends on the unknown distribution P. It is shown here that if the sequences An and θn 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 An* and θn* are analogs of An and θn computed from a sample from Pθn, the empirical processes n1/2(An−Aθn) and n1/2(An*−Aθn*) 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.
LA - eng
KW - copula; goodness-of-fit test; Monte Carlo simulation; parametric bootstrap; P-values; semiparametric estimation; -values
UR - http://eudml.org/doc/78005
ER -

References

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