Suppose that an observation $x \in (-1,1)$ comes from a sample model with a parameter $\theta$, with density function:
$$ f(x\mid\theta) = \begin{cases} \theta\ &\text{if}& -1 < x < 0\\ 1 - \theta\ &\text{if}& \ 0 \leqslant x < 1 \end{cases}\ $$
Consider the decision rule $δ_0(x) = (1 − x)/2$ and the quadratic loss function $L(\theta , d) = (\theta - d)^2$.
Suppose that the prior distribution for $\theta$ is uniform over the interval $(0,1)$.
a) Find the sufficient statistic of $\theta$.
b) Using the Rao-Blackwell Theorem, find a decision rule that dominates $δ_0$, which is $E(δ_0(X) \mid T=t)$
For part a), I was trying to factorize $f(x\mid\theta)$ using the indicator function but couldn't get anything. Could someone help me out?
Thank you!