Suppose that an observation $x \in (-1,1)$ comes from a sample model with a parameter $\theta$, with density function:
$$ f(x\mid\theta) = \begin{cases} \theta\ if -1 < x < 0\\ 1 - \theta\ if\ 0 \leqslant x < 1 \end{cases}\ $$
a) Consider the decision rule $δ_0(x) = (1 − x)/2$. Find the frequentist risk $R(θ; δ_0)$ associated with this decision rule.
I got $\theta^3 -\theta^2 +0.5/6$
b) Suppose that the prior distribution for $\theta$ is uniform over the interval $(0,1)$. Find a Bayes estimator associated with this prior distribution, under the quadratic loss function $L(\theta , d) = (\theta - d)^2$.
c) Using the Rao-Blackwell Theorem, find a decision rule that dominates $δ_0$ and verify explicitly that this new rule dominates $δ_0$.
I have no idea how to deal with question c). For part b), I'm wondering if this Bayes estimator is $E(\theta\mid x) = (2\theta - 1)^2 +1 $
Please let me know your thoughts. Thank you!