Compromise Pitman estimator of location

Consider the modelY = μ+ σ · E, μ ∈ R, σ ∈ R+,where E follows a known distribution F. The parameter μ is called the location parameter. Pit-man (1939) introduced a method to determine an optimal equivariant estimator of μ, when thedistribution F is continuous, but otherwise of any form. This method uses a conditioning on anancillary statistic to minimize the mean square error.In this talk we present a way to use Pitman’s method when the error distribution is unknown.Let F be a given set of distributions. We propose to choose one or several members of this set,F1, . . . , Fm, and construct the final estimator by computing a weighted mean of the Pitman esti-mators associated to each chosen distribution. The weights will depend on the observed likelihoodof the models.The question of the choice of the distribution within the set F , and the value of m, the numberof distributions to be taken, will be studied trough the asymptotic behavior of the final compromiseestimator. A minimax approach on a distance between distributions will be presented. ReferencesE. J. G. Pitman (1939). The Estimation of the Location and Scale Parameters of a ContinuousPopulation of any Given Form. Biometrika, 30, 3/4, 391–421.