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 moderate-deviations 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.
Developing p-values: a Bayesian-frequentist convergence
Université de PARIS - DAUPHINE
Place du Maréchal de Lattre De Tassigny - 75775 PARIS CEDEX 16 - FRANCE
Téléphone : +33 (0)1 44-05-49-23 - fax : +33 (0)1 44-05-45-99