I have some pdf $f(x|\theta) = (\theta/2) ^ {|x|} \cdot (1 - \theta) ^ {(1 - |x|)}$ where $x = -1, 0, 1$, and $0 \le \theta \le 1$.
I found the MLE of theta by denoting $n_{-1}$ the number of realizations where $x=-1$, $n_{0}$ the number of realizations where $x=0$ and $n_{1}$ the number of realizations where $x=1$. $n=n_{-1}+n_{0}+n_{1}$ is the size of the sample. The joint density of the sample I obtained was
$$f(X\mid \theta) = \Big[(\theta/2) ^ {|-1|} \cdot (1 - \theta) ^ {(1 - |-1|)}\Big]^{n_{-1}}\cdot \Big[(\theta/2) ^ {|0|} \cdot (1 - \theta) ^ {(1 - |0|)}\Big]^{n_{0}} \cdot \Big[(\theta/2) ^ {|1|} \cdot (1 - \theta) ^ {(1 - |1|)}\Big]^{n_{1}}$$
$$=(\theta/2) ^ {n_{-1}} \cdot (1 - \theta) ^ {n_{0}} \cdot (\theta/2) ^ {n_{1}} = (\theta/2) ^ {n-n_{0}} \cdot (1 - \theta) ^ {n_{0}}$$ and the log-likelihood
$$\ln L = (n-n_{0})\ln \theta + (n-n_{0})\ln2+ n_{0}\ln (1-\theta)$$ with the MLE being $\hat{\theta}$ =($n$-$n_{0}$) \ ($2n_{0}$-$n$)
Now I need to define the estimator T(X)= 2 if X=1 and zero otherwise. Find a better estimator than T(X) and prove that it is better. I am really confused on how to approach this problem.
Please do not use the internet to find answers to my assignment. These assignments are meant to test your knowledge and prepare you for the final exam. Because of this post, everyone will now redo all assignments again with new assessment pieces to make sure there is no plagiarism, I will discuss this in our lecture on Tuesday.