# Rao-Blackwell for Minimum-Variance Unbiased Estimator

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?

• By Factorisation theorem, $|X|$ is sufficient (and complete since this is exponential family). Apr 28 '19 at 21:41

Hint: Write out the sampling distribution for the statistic $$|X|$$. It is an extremely simple and well-known distributional form. Once you have found this, finding the MVUE should not present much problem.