Timeline for Monotone likelihood ratio of a family of densities with compact support
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Aug 14, 2023 at 1:58 | comment | added | japalmer | So the new conjecture is that it holds with the additional condition that $g^{(3)}(x) \ge 0$ for $x > 0$. | |
| Aug 14, 2023 at 1:38 | comment | added | japalmer | Yes, I checked it by inverting the convex function. It's strange because I'm pretty sure I can prove that if $g$ is concave or convex and $a(\phi)$ is montonic, then $b(\theta) = \int_0^{\pi/2} a(\phi) p(\phi;\theta) d\phi$ is monotonic. (reversed for concave $g$). I thought this was a stronger property given a theorem by Karlin on TP(3) kernels having that property, which is stronger than the SR(2) property, which is equivalent to the monotone likelihood ratio. | |
| Aug 13, 2023 at 16:58 | comment | added | Iosif Pinelis | @japalmer : The "concave" part of your conjecture has now considered as well. | |
| Aug 13, 2023 at 16:57 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Aug 13, 2023 at 16:51 | vote | accept | japalmer | ||
| Aug 13, 2023 at 16:50 | comment | added | japalmer | I guess the inverse of that function also disproves the concave case. It must need a 3rd derivative condition. | |
| Aug 13, 2023 at 16:27 | comment | added | japalmer | Wow, that's not good. I thought it applied in the convex case as well. But I'm pretty sure I can show that it holds for concave $g$. Do you have a counter-example for that too? | |
| Aug 13, 2023 at 16:03 | history | answered | Iosif Pinelis | CC BY-SA 4.0 |