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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.

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    $\begingroup$ Is this homework? $\endgroup$
    – Glen_b
    Sep 12 at 0:05
  • $\begingroup$ Do you just need to give such an example? // Is this homework? $\endgroup$
    – Dave
    Sep 12 at 0:16
  • $\begingroup$ 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. $\endgroup$
    – Ivor
    Sep 12 at 1:06
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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.

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  • $\begingroup$ ... as a hint, this is a good one (+1). Hopefully it will be enough. $\endgroup$
    – jbowman
    Sep 12 at 4:30

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