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

Kucerovsky Dan, Marchand Eric, Amir Payandeh and Strawderman William E.
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Kucerovsky Dan: University of New Brunswick, Department of Mathematics and Statistics, CANADA E3B 5A3
Strawderman William E.: Rutgers University, Department of Statistics, Piscataway, N.J., U.S.A.

Statistics & Risk Modeling, 2009, vol. 27, issue 02, 145-168

Abstract: 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.

Keywords: absolute value error loss; admissibility; Bayes estimators; Fourier transform; Fredholm integral equations (search for similar items in EconPapers)
Date: 2009
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DOI: 10.1524/stnd.2009.1026

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