# Unbiased estimation of square in presence of Gaussian noise

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.

• I don't have a solution, but a mathematical setup for your problem. Let's denote the estimator as a function $f(\cdot)$. You observe one sample of $Y \sim \mathcal{N}(x,\sigma^2)$. We want to find $f$ that satisfies unbiasedness $E[f(Y)]=x^2$ and non-negativity $P(f(Y)<0)=0$. – Atul Ingle Jul 7 '17 at 20:46
• @ SRT, did you get anywhere with this question? – Atul Ingle Aug 6 '17 at 14:50
• Perhaps we should move this to stats stackexchange? – Atul Ingle Aug 6 '17 at 15:03
• Hi @AtulIngle, thanks for your interest. No, I didn't get anywhere with this question. I am open to move this to stats stackexchange, but unclear how to do it. – SRT Aug 10 '17 at 21:17