I have a single scalar observation, $y$, of a scalar parameter $x$ in presence of additive Gaussian noise. That is $y = x + n$ where $n$ is the Gaussian noise. The variance $n$ is known to be $\sigma^2$.
The problem: estimate $x^2$ from the observation $y$.
It appears that I can have an unbiased estimator which is $y^2 - \sigma^2$. However, this estimate can be negative whereas $x^2$ cannot be negative. Therefore, I can amend my estimate to be $\max(y^2-\sigma^2, 0)$. However, this estimate is no longer unbiased.
The question is: what is a good (preferably unbiased) estimate of $x^2$ from $y$ that is always non-negative.