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Is there a name for the following process?

Say I have an absolutely continuous probability density function $f$ with compact support, and I take $k$ independent samples $x_1,\dots,x_k$ from $f$. Then, I let $x^*$ be the sample for which $f(x_i)$ is the largest, i.e. $x^* = \arg \max_i f(x_i)$. My question is, is there a name for this process, and can anything be said about the distribution of $x^*$? This basically has the effect of emphasizing the modes of $f$ more aggressively.

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  • $\begingroup$ "Subsampled estimate of the mode." But in your problem is $f$ known? Otherwise, how would you compute $x^*$? $\endgroup$ – David G. Stork Apr 8 '16 at 23:39
  • $\begingroup$ Yes, $f$ is known. I'm interested in describing the distribution of $x^*$ in terms of $f$. $\endgroup$ – Tom Solberg Apr 8 '16 at 23:51
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    $\begingroup$ As $k\to\infty$, the distribution of $x^*$ will obviously be concentrated near the maximum of $f$ (assuming for simplicity that $f$ has a unique global max), as you already pointed out yourself. What else do you hope to be able to say in this generality? $\endgroup$ – Christian Remling Apr 9 '16 at 0:22
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If your $k$ samples are taken with replacement, then your method is a Bootstrap estimate of the mode. In the limit of a large number of such samples, the Bootstrap estimate coincides with the true mode.

Bootstrapping is useful when seeking the variance of a statistic such as the mode, as in your case.

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