- #1
Boot20
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Homework Statement
[itex]W(\theta - \delta)[/itex] the loss function.
[itex]\theta[/itex] the true parameter.
[itex]\delta[/itex] an estimator of [itex]\theta[/itex]
W a smooth, non-negative, symmetric, convex function.
[itex]p(\theta | x)[/itex] the posterior density of the parameter [itex]\theta[/itex].
Prove that, for normal posterior density [itex]p(\theta | x)[/itex] and under the loss function specified, the estimator that minimize the expected loss is the posterior mean.
Homework Equations
The expected loss can be written as
[itex]U(\delta) = E[W(\theta - \delta)] = \int W(\theta - \delta) p(\theta | x)d\theta[/itex]
The Attempt at a Solution
I really have no idea where to start. I suspect that [itex]U(\delta) [/itex] is also convex but I know no way to prove it.