# What is meant by optimal estimator and how to determine optimality?

Considering an estimation problem of estimating a scalar deterministic parameter $a$ from the observations $y$ which are corrupted by randomvariable $w$. The observations are $y[n] = a + w[n]$

Least Squares estimator can be used to estimate $a$ when $w$ is a White Gaussian Random Variable. This estimation method is known to be optimal. Why?

What if $w$ is from Poisson Distribtuion or some other non-gaussian, then would the estimator for $a$ be better or worse than the one found using $w$ as Gaussian r.v?

You can generate tons of estimators by defining different cost functions. Other popular cost functions are l1-norm ($\sum |x_i - y_i|$), likelihood ($P(y|x)$).
• Thanks for your comment. There is no distribution for the parameter a, it is a deterministic value and I have obtained it randomly from a Gaussian distribution. I am checking which noise distribution gives the estimate of the parameter that is the "best" or closest to the known value. This is the reason I ask if there is any theory which says how to determine which estimator is the best. – SKM Aug 3 '17 at 21:55