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On the Bayesianity of maximum likelihood estimators of restricted location parameters under absolute value error loss. (English) Zbl 1274.62169

Summary: We investigate the potential Bayesianity of maximum likelihood estimators (MLE), under absolute value error loss, for estimating the location parameter \(\theta\) of symmetric and unimodal density functions in the presence of (i) a lower (or upper) bounded constraint, and (ii) an interval constraint, for \(\theta\). With these problems being expressed in terms of integral equations, we establish for logconcave densities: the general 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), our key tool resides in a correspondence with a Riemann-Hilbert problem, while in (ii) we use Fredholm’s technique. We demonstrate that logconcavity is critical with a class of counterexamples. Finally, various other remarks, illustrations and numerical evaluations are provided.

MSC:

62F10 Point estimation
62F30 Parametric inference under constraints
62C10 Bayesian problems; characterization of Bayes procedures
35Q15 Riemann-Hilbert problems in context of PDEs
45B05 Fredholm integral equations
42A99 Harmonic analysis in one variable
62C15 Admissibility in statistical decision theory
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