Three strings of inequalities among six Bayes estimators

Abstract

This is the supplemental file of the paper.Theorem 1. Assume the prior satisfies some regularity conditions such that the posterior expectations involved in the definitions of the six Bayes estimators exist. Then for Θ = (0, 1), there is a string of inequalities among the six Bayes

The Bayesian Posterior Estimators under Six Loss Functions for Unrestricted and Restricted Parameter Spaces

“…The answer to this question is no! The numerical simulations of the smallest PELs exemplify this fact (see [36]…”

“…For each one of the six loss functions, we can find a corresponding Bayesian estimator, which minimizes the corresponding posterior expected loss. Among the six Bayesian estimators, there exist three strings of inequalities summarized in Theorem 1 (see also Theorem 1 in [36]). However, a string of inequalities among the six smallest PELs does not exist.…”

“…For the six loss functions, we have the corresponding six Bayesian estimators δ π w2 x ðÞ , δ π pl x ðÞ , δ π s x ðÞ , δ π p x ðÞ , δ π 2 x ðÞ ,andδ π z x ðÞ . Interestingly, for the six Bayesian estimators, we discover three strings of inequalities which are summarized in Theorem 1 (see Theorem 1 in [36]). To our surprise, an order between the two Bayesian estimators δ π w2 x ðÞ and δ π pl x ðÞ on Θ ¼ 0; 1 ðÞ does not exist.…”

“…In this section, we compare the six Bayesian estimators δ π w2 x ðÞ , δ π pl x ðÞ , δ π s x ðÞ , δ π p x ðÞ , δ π 2 x ðÞ , and δ π z x ðÞ . The domains of the loss functions, the six Bayesian estimators, the PELs, and the smallest PELs are summarized in Table 2 (see Table 1 in [36]). The six PELs are PEWSEL, PEPLL, PESL, PEPL, PESEL, and PEZL.…”

“…The proof of Theorem 1 exploits a key, important, and unified tool, the covariance inequality (see Theorem 4.7.9 (p. 192) in [16]), and the proof can be found in the supplement of [36].…”