Bayesian inference on a squared quantity

It is here derived the Bayesian estimator of the noncentrality parameter of the noncentral chi-square distribution. The corresponding frequentist estimator, based on the method of moments, is also derived and its performance is compared with the Bayesian one. The Bayesian estimator is obtained through an analytical derivation which provides insight into the way the estimator works. Reference is also made here to a previously published work on a similar subject by Filippo Attivissimo, Nicola Giaquinto and Mario Savino (“A Bayesian paradox and its impact on the GUM approach to uncertainty,” Measurement 45 (2012), pp. 2194-2202) in order to resolve the paradox there presented. Some defects of the analysis performed in the referenced work are identified and carefully examined. The superiority of the Bayesian estimator is demonstrated although achieved at the price of a greater complexity.