A loss function investigated by Zellner (1986) is the LINEX (LINear-EXponential) loss, a loss function that can handle asymmetries in a smooth way. The LINEX loss is given by
L(8,a) = ec(a-θ) - c(a - 8) - 1,
where c is a positive constant. As the constant c varies, the loss function varies from very asymmetric to almost symmetric.
(a) For c = .2, .5, 1, plot L(θ, a) as a function of α - θ.
(b) If X ~ F(x|θ), show that the Bayes estimator of 8, using a prior π, is given by

(c) Let X1,..., Xn be iid n(θ,σ2), where σ2 is known, and suppose that θ has the noninformative prior π(θ) = 1. Show that the Bayes estimator versus LINEX loss is given by δB() = -(cσ2/(2n)).
(d) Calculate the posterior expected loss for δB(X) and X using LINEX loss.
(e) Calculate the posterior expected loss for δB(X) and X using squared error loss.

Transcribed Image Text:

6" (X) = log E(e-c|X).