# Choosing a sample based on where the density function is highest

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.

• "Subsampled estimate of the mode." But in your problem is $f$ known? Otherwise, how would you compute $x^*$? – David G. Stork Apr 8 '16 at 23:39
• Yes, $f$ is known. I'm interested in describing the distribution of $x^*$ in terms of $f$. – Tom Solberg Apr 8 '16 at 23:51
• 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? – Christian Remling Apr 9 '16 at 0:22

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.