Is there any unbiased estimator for the regression coefficient $\beta \in \mathbb{R}^p$, p >> 1, where $$ y_k = x_k^T\beta + \epsilon \in \mathbb{R}? $$ Note that $x_k \in \mathbb{R}^p$ and $\epsilon \sim \mathcal{N}(0, \sigma^2)$, for some $\sigma > 0$.

  • $\begingroup$ It would be more standard to use $p$ instead of $n$ for the dimension and then use $n$ to denote sample size; consider updating your post. With this new notation, how large is $p$ relative to $n$? If $p<n$, you can use OLS. $\endgroup$ May 12 at 6:03
  • $\begingroup$ It is written as it is. In your notation, p = n, n = 1. $\endgroup$
    – user808843
    May 12 at 15:47
  • $\begingroup$ Would the gauss-markov-theorem not apply? $\endgroup$
    – Dave
    May 12 at 16:00
  • $\begingroup$ How large is $n$? Is it larger or smaller than $p$? If $p<n$, you can use OLS. $\endgroup$ May 12 at 20:06
  • $\begingroup$ Please read the question description as it is. I have specified that p >> n = 1. I am less confident on the existence of unbiased estimator in this setting. $\endgroup$
    – user808843
    May 12 at 21:18


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