Let $X$ be a random variable having pmf:
\begin{array}{ll} p(x)=2 \theta \ \ \ \text{if} \ x=-1 \\ p(x)=\theta^2 \ \ \text{if} \ x=0 \\ p(x)=1-2\theta-\theta^2 \ \ \text{if} \ x=1\\ \\ \end{array} where $\theta \in [0,\sqrt{2}-1]$.
Show that there is one and only one unbiased estimator of $(\theta+1)^2$ based on a single observation.
I have been able to construct an unbiased estimator but I am failing to prove its uniqueness.
Define $I=1 \ \text{if} \ \ {x=0}\ \text{or} \ {x=-1}$ and $0$ otherwise. $E(1+I)=1+P(X=0)+P(X=-1)=(\theta+1)^2$ But how to show that it is the only unbiased estimator?
I think if we can prove that $1+I$ is UMVUE then I think uniqueness is implied.
Suppose that $f(X)$ is an unbiased estimator of $(\theta+1)^2$. Then for any $\theta\in[0,\,\sqrt{2}-1]$ $$ (\theta+1)^2=\mathbb E[f(X)]=2\theta f(-1)+ \theta^2 f(0)+(1-2\theta-\theta^2)f(1). $$ Rewrite this as following: for any $\theta\in[0,\,\sqrt{2}-1]$, $$ \theta^2(f(0)-f(1)-1)+2\theta(f(-1)-f(1)-1)+(f(1)-1)=0. $$ By Fundamental theorem of algebra, this polynomial can have infinite number of roots only if all its coefficients are zero. So, $f(1)=1$, $f(-1)=f(0)=2$ give us the unique unbiased estimator $f(X)$.