Let $X$ be an observation from a distribution with probability mass function:$f(x;\theta) = \left(\frac{\theta}{2}\right)^{|x|}(1-\theta)^{1-|x|}I_{\{-1,0,1\}}(x), \, \theta \in (0,1).$ Use Rao-Blackwell theorem to find the Minimum-Variance Unbiased Estimator (MVUE).
To begin with, the PMF can also be written as $$ f(x;\theta) = I_{\{-1,0,1\}}(x) (1-\theta)\exp \left\{|x| \ln\left(\frac{\theta}{2(1-\theta)}\right)\right\} $$ and therefore $f$ belongs to the Exponential Family. According to Pitman-Koopman, there exists a sufficient function for the parameter $\theta$.
Now, from R-B: $W = E[R = |x| \big| T]$ (where $R = |x|$ is an Unbiased estimator of $\theta$ and $T$ is a sufficient and complete function for $\theta$).
The problem I face is that I don't get how we construct a $T$ which is sufficient and complete for $\theta$ and how does one use it to find an explicit formula for $W$. Can someone provide me with these two steps for finding the MVUE?
