Summary
Motivated by attractive features of inferences based on heavy-tailed distributions, we develop Bayesian inference for the location parameter of a family of location-scaled distributions, which are shifted and scaled Studentt and normal distributions. The family of prior distributions utilized to the unknown location and squared-scale consists of independent Studentt and inverted gamma distributions. When observations are sampled from at distribution, the likelihood estimators are intractable. Thus, we use MML (modified maximum likelihood) estimators instead to develop Bayesian inference. Bayesian estimators are defined by modes of posterior densities and called HPD (highest posterior density) estimators. The proposed estimators, especially the estimators arising from at distribution, are clearly superior. They adjust automatically to the sample dispersion, ignore inconsistent information, and are rather insensitive to outliers. The posterior density of the location parameter may be bimodal. Posterior bimodality indicates some conflicts between the two sources of information: the sample and the prior knowledge. We also report the results of simulation studies that point toward overall superiority of the proposed estimators.
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Bian, G. Bayesian inference in location-scale distributions with independent bivariate priors. Test 6, 137–157 (1997). https://doi.org/10.1007/BF02564431
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DOI: https://doi.org/10.1007/BF02564431