Timeline for What is "the shortest half of the data"?
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| Dec 29, 2021 at 21:17 | comment | added | whuber♦ | @Alexis Imagine a distribution with many modes. One way to get one begins with a dataset: convolve that with a narrow-bandwidth kernel. The result is s a lot of narrow peaks located near the data values. The shortest-length region is a union of small intervals around each data point--and we have come full circle back to my original example! Some distributions have infinitely many modes. Take the density proportional to $2+\sin(1/|x|)$ for $-\pi\lt x\lt \pi,$ for instance. Its shortest-length half has infinitely many connected components. | |
| Dec 29, 2021 at 21:03 | comment | added | Alexis | @whuber Ah, thank you. Regarding the possibly "very disconnected" set, is there an intuitive graphical representation of this? My mind wants to jump to a set of range(s) demarcating the most massive ranges of probability density… something like that? | |
| Dec 29, 2021 at 20:35 | comment | added | whuber♦ | @Alexis If contiguity were not required, you could generate examples by forming balls of radius $\epsilon$ around each of the data values and taking the union of half of them. The limit of the total length as $\epsilon\to 0$ is zero, no matter how you choose the subsets, so nothing is accomplished. For continuous distributions, though, with density function $f,$ there will be some maximal threshold $t$ for which the total probability of the set $x:f(x)\gt t$ is $1/2$ or greater. This set has minimal length among all sets of probability $1/2$ or greater--and can be very disconnected. | |
| Dec 6, 2014 at 16:32 | comment | added | Alexis | Do those halves need to be contiguous? I think the answer is yes, but want to be sure. | |
| Sep 10, 2014 at 22:58 | vote | accept | Remi.b | ||
| Sep 10, 2014 at 16:13 | history | answered | Greg Snow | CC BY-SA 3.0 |