# Nonlinear unbiased estimator with strictly smaller variance than OLS estimator

I want to find the following example: Construct a linear model $$y=X\beta + \epsilon$$, where $$\epsilon$$ is mean zero uncorrelated homoscedestic non-Gaussian noise and $$X$$ is deterministic, such that there exists a nonlinear unbiased estimator that has a strictly smaller variance than the least square estimator.

I have no idea how to proceed.

• Is this homework? Sep 12 at 0:05
• Do you just need to give such an example? // Is this homework?
– Dave
Sep 12 at 0:16
• It's a question from a class I'm auditing for my job. I'm not a statistician so I can't figure this out myself. Any pointer would be helpful.
– Ivor
Sep 12 at 1:06

I would start with the one-sample case. Given the trivial linear model $$y=\mu+\epsilon$$, where $$\epsilon$$ is mean zero uncorrelated homoscedestic noise, the OLS estimator is just the sample average. What are some non-linear estimators of $$\mu$$? Can you choose $$\epsilon$$ so one of them is better than the sample average? If so, can you expand this to where $$\mu=X\beta$$ is non-trivial.
One choice for a non-linear estimator would be the median, so you need to find a distribution for $$\epsilon$$ where the median is unbiased for the mean but has strictly smaller variance than the mean.