On the Bayesianity of maximum likelihood estimators of restricted location parameters under absolute value error loss

We investigate the potential Bayesianity of maximum likelihood estimators (MLE), under absolute value error loss, for estimating the location parameter θ of symmetric and unimodal density functions in the presence of (i) a lower (or upper) bounded constraint, and (ii) an interval constraint, for θ. With these problems being expressed in terms of integral equations, we establish for logconcave densities: the generalized Bayesianity of the MLE in (i); and the proper Bayesianity and admissibility of the MLE in (ii) which extends the normal model result of Iwasa and Moritani. In (i), a key feature concerns a correspondence with a Riemann–Hilbert problem, while in (ii) we use Fredholm´s technique and a contraction mapping argument. We demonstrate that logconcavity is a critical condition with sufficient conditions for non-Bayesianity and, accordingly, with a class of counterexamples. Note that the Bayesianity of the MLE under absolute value loss in the restricted location parameter case is in marked counterdistinction to that under quadratic loss, where, typically, a generalized Bayes estimator must be a smooth function. Finally, various other remarks, illustrations and numerical evaluations are provided.