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I'm trying to understand what is the minimum of the following function,

$$ f(\mu) = \frac{1}{n}\sum_{i=1}^n F\left(\frac{\pi(i)}{n}\right) (x_i - \mu )^2 $$

where $F$ is a step function that assign $1$ to any $t \in [0, \gamma]$ and $0$ in $(\gamma, 1]$, $\pi$ is a permutation that sorts the $i$s based on the ascending ordering of the losses $( x_i - \mu )^2$ and $\mu$ and $x_1,\ldots,x_n$ are real numbers. I was thinking that perhaps the minumum of $f$ is achieved by the trimmed mean, namely: sort the $x_i$s discard the first $(1-\gamma)/2$ and the last $(1-\gamma)/2$ of the $x_i$ and compute the arithmetic mean of the remaing points. However, it isn't as a simple numeric example shows.

Any ideas or maybe suggestions (perhaps it is a known problem)? Many thanks in advance.

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  • $\begingroup$ Have you worked through some examples? What have you found there? Did all the examples you tried match your conjecture? If you haven't tried that yet, I recommend you try that before asking here. I think that will help you figure out what is going on, and correct your conjecture. $\endgroup$
    – D.W.
    May 24, 2020 at 17:20
  • $\begingroup$ Many thanks indeed it is not the trimmed mean. Unfortunately, I can't figure out other conjectures for the minimum. $\endgroup$
    – Andrea
    May 24, 2020 at 17:35
  • $\begingroup$ Try figuring out what happens for small $n$, say $n=2,3,4$. $\endgroup$ May 24, 2020 at 18:30
  • $\begingroup$ Thanks Yuval, it has been useful to look at small examples. Thanks also to orlp's answer, I think I managed to prove that the minimum is the mean of the subsequence (of continguous elements) with smallest variance. $\endgroup$
    – Andrea
    May 25, 2020 at 10:22

1 Answer 1

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Your loss function is proportional to the $\gamma n$ smallest square differences with respect to $\mu$.

This is minimized by sorting your $x_i$ and choosing the mean of the contiguous substring $x_k, x_{k+1}, \dots, x_{k+\gamma n-1}$with the smallest variance.

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  • $\begingroup$ Thanks a lot for your answer. $\endgroup$
    – Andrea
    May 25, 2020 at 10:26

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