Cahiers du CEREMADE 

Unité
Mixte de Recherche du C.N.R.S. N°7534 

Abstract : Composite null hypotheses are often
assessed by constructing a plausible measure of departure of data
from expectation under the hypothesis and then seeking a modified
measure that is distribution free of any nuisance parameters and
thereby providing the corresponding $p$value. Such $p$values
for a composite null hypothesis have had extensive attention in
the Bayesian literature, with some preference shown for two
versions designated $p_{
m ppost}$ and $p_{
m cpred}$. And it
has been indicated that certain candidate $p$values can be
upgraded to the preceding preferred $p$values by the parametric
bootstrap. From another direction, recent likelihood theory gives
a factorization of a statistical model into a marginal density for
a full dimensional ancillary and a conditional density for the
maximum likelihood variable. Here, for any given initial trial or
test statistic that provides a location for a data point we
develop: a special version of the Bayesian $p_{
m cpred}$; an
ancillary based $p$value designated $p_{
m anc}$; and a
bootstrap based $p$value designated $p_{
m bs}$. We then show
under moderate regularity that these are equivalent to the third
order and have uniqueness as a determination of the statistical
location of the data point, as of course derived from the initial
location measure. We also show that these $p$values have a
uniform distribution to third order, as based on calculations in
the moderatedeviations region. This gives three very different
routes to a unique $p$value corresponding to the initial trial
measure of location and composite null hypothesis. We thus have
great flexibility for the derivation of $p$values, from a
convergence of Bayesian and frequentist points of view. Some
examples are given to indicate the ease and flexibility of the
approach. 





200530 

06072005 

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