Timeline for Sign Regularity of a Density Kernel with Convexity Properties
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 16, 2023 at 6:21 | comment | added | japalmer | Not true for completely Bernstein functions (operator monotone functions) in general either. Must depend as well on the relative magnitudes of the first and second derivatives or something. I need to find an integral representation that applies to the listed functions e.g., and has a kernel with MLRP. | |
| Aug 16, 2023 at 5:37 | comment | added | japalmer | Actually it's not true in general even for Bernstein $g$. There must be some way to characterize the functions this is true for other than the property itself. | |
| Aug 16, 2023 at 3:48 | comment | added | japalmer | I use Maple, but I can easily check derivatives and plots. | |
| Aug 16, 2023 at 3:26 | comment | added | japalmer | I think it will be possible using the Levy-Khinchine representation of Bernstein functions. The $m$th integral of the Levy-Khinchine kernel can be calculated as well. Proving that these kernels have monotone likelihood ratio, and then that the scale mixing density also has MLRP should do it. | |
| Aug 15, 2023 at 20:35 | comment | added | Iosif Pinelis | @japalmer : This will probably be hard to prove for a general class of functions. You may want to try different ones using my Mathematica notebook. | |
| Aug 15, 2023 at 17:05 | comment | added | japalmer | Last resort conjecture: $g$ is a Bernstein function in the concave case, or $g$ is an $m$th integral of a Bernstein function, or a scale mixture of $\cosh(x)$ in the convex case. | |
| Aug 15, 2023 at 16:21 | comment | added | japalmer | Actually not absolutely monotonic, as $|x|^p$ works for $p>2$. Maybe just univalent derivatives of all orders. | |
| Aug 15, 2023 at 16:19 | vote | accept | japalmer | ||
| Aug 15, 2023 at 16:19 | comment | added | japalmer | Damn. That function apparently with appropriate exponent seems to work for any finite number of derivative conditions. I guess it requires absolutely monotonic or Bernstein functions. I'll work on it before addressing it here again. | |
| Aug 15, 2023 at 15:32 | history | answered | Iosif Pinelis | CC BY-SA 4.0 |