Gaussian distribution variance estimation

Question

It's well known if I have a process generating normally distribuited data, I can estimate the parameters of the gaussian function: $$N(\mu,\sigma)=\frac{1}{\sqrt{2\pi\sigma^2}}\exp\left(-\frac{1}{2}\frac{(x-\mu)^2}{\sigma^2}\right)$$ using the data generated by that process. The precision of the estimation increases increasing the amount of data. Supposing I know the parameter $\mu$. Is it possible to have some information about $\sigma$ with only one realization of the process that is with only one measurement? Thanks.

Answer 1

Supposing you know the mean is $\mu$ and your one measured value is $x$, it's pretty easy to show that the maximum-likelihood estimate for $\sigma$ is $|\mu-x|$. However, this estimator will usually guess too low. An unbiased estimator would be $|\mu-x| \sqrt{\pi/2}$.

An unbiased, maximum-likelihood estimator for the variance $\sigma^2$ is given simply by $(\mu-x)^2$.

One trial is really not enough to get a good idea of the distribution.