Testing Un-Separated Hypotheses by Estimating a Distance

Jean-Bernard Salomond (1)

1 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées

In this paper we propose a Bayesian answer to testing problems when the hypotheses are not well separated. The idea of the method is to study the posterior distribution of a discrepancy measure between the parameter and the model we want to test for. This is shown to be equivalent to a modification of the testing loss. An advantage of this approach is that it can easily be adapted to complex hypotheses testing which are in general difficult to test for. Asymp-totic properties of the test can be derived from the asymptotic behaviour of the posterior distribution of the discrepancy measure, and gives insight on possible calibrations. In addition one can derive separation rates for testing, which ensure the asymptotic frequentist optimality of our procedures.

Mots clés: goodness-of-fit; asymptotic properties of tests; nonparametric inference; hypothesis testing; Bayesian inference; monotonicity

Domaines: Statistiques [math.ST]

Jean-Bernard Salomond. Testing Un-Separated Hypotheses by Estimating a Distance. Bayesian Analysis, 2017, ⟨10.1214/17-BA1059⟩. ⟨hal-01585786⟩