G-2005-51
Validity of the Parametric Bootstrap for Goodness-of-Fit Testing in Semiparametric Models
and BibTeX reference
In testing that a particular distribution belongs to a parameterized family
, one
often compares a non-parametric estimate
of
, with a member
of
. In most
cases, the limiting distribution of goodness-of-fit statistics based on
depends on the unknown distribution
. It is shown here that if the sequence (
) of estimators is regular in some sense, then the parametric bootstrap approach is valid, i.e., if
and
are analogs of
and
calculated from a bootstrap sample from
, then the empirical processes
and
converge jointly
to independent and identically distributed limits. These results are used to establish
the validity of the parametric bootstrap method when testing the goodness-of-fit
of parametric families of multivariate distributions and copulas. Two types of tests
are considered: those based on a distance between an empirical multivariate distribution
function or copula and its parametric estimation under the null hypothesis, and
those based on a distance between empirical and parametric estimations of univariate
pseudo-observations obtained via a probability integral transformation. For situations
in which the multivariate distribution function or its probability integral transformation
cannot be obtained in closed form, a two-level parametric bootstrap is developed
and its validity is established. As an illustration, the one- and two-level parametric
bootstrap methodology is detailed in the special case of a goodness-of-fit test statistic
for copula models based on a Cram´er–von Mises type functional of the empirical copula
process.
Published July 2005 , 42 pages