The Bayes rule of the variance parameter of the hierarchical normal and inverse gamma model under Stein's loss

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

“…Model (a) (hierarchical normal and inverse gamma model). This hierarchical model has been investigated by [10,16,17]. Suppose that we observe X 1 ,X 2 , …,X n from the hierarchical normal and inverse gamma model:…”

“…The gamma prior G α; β ðÞ is a conjugate prior for the Poisson model, so that the posterior distribution of θ is also a gamma distribution. The hierarchical Poisson and gamma model (10) has been considered in Exercise 4.32 (p. 196) of [4]. It has been shown that the marginal distribution of X is a negative binomial distribution if α is a positive integer.…”

“…However, they did not provide any Bayesian posterior estimator of θ. Moreover, the normal distribution with a normal-inverse-gamma prior which assumes that μ is unknown is more realistic than the normal distribution with an inverse gamma prior investigated by [10] which assumes that μ is known.…”

“…A Bayesian estimator minimizes some posterior expected loss (PEL) function. We confine our interests to six loss functions in this chapter: the squared error loss function (well known), the weighted squared error loss function ( [1], p. 78), Stein's loss function [2][3][4][5][6][7][8][9][10], the power-power loss function [11], the power-log loss function [12], and Zhang's loss function [13]. It is worthy to note that among the six loss functions, the first and second loss functions are defined on Θ ¼À ∞; ∞ ðÞ , and they penalize overestimation and underestimation equally.…”

“…For Θ ¼ 0; ∞ ðÞ , the positive restricted parameter space, where 0 is a natural lower bound and the estimation problem is not symmetric, we should not choose the squared error loss function and the weighted squared error loss function but choose a loss function which can penalize gross overestimation and gross underestimation equally, that is, an action a will suffer an infinite loss when it tends to 0 or ∞. Stein's loss function owns this property, and thus it is recommended for Θ ¼ 0; ∞ ðÞ by many researchers (e.g., see [2][3][4][5][6][7][8][9][10]). Moreover, [11] proposes the power-power loss function which not only penalizes gross overestimation and gross underestimation equally but also has balanced convergence rates or penalties for its argument too large and too small.…”