Timeline for Monotone likelihood ratio of a family of densities with convexity property
Current License: CC BY-SA 4.0
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| Aug 15, 2023 at 14:37 | vote | accept | japalmer | ||
| Aug 15, 2023 at 14:26 | answer | added | Iosif Pinelis | timeline score: 1 | |
| Aug 15, 2023 at 13:51 | comment | added | japalmer | Classes of functions, analogous to multiply montone and (multiply) absolutely monotone functions can be defined, where the function has $n$ positive derivatives, or is concave, with $n$ alternating sign derivatives, basically $n$-times Bernstein. The functions described here would be 3-times absolutely monotonic, or Bernstein, or possibly 4-times, in $x^2$. | |
| Aug 15, 2023 at 13:46 | comment | added | japalmer | As I imagine the combinatorial and tedious proof going, it might require that the $g^{(4)}(x)$ be univalent on $(0,\infty)$ to guarantee that the last of the three transformations has the MLRP. So perhaps a counter-example could be found. | |
| Aug 15, 2023 at 13:40 | comment | added | japalmer | @IosifPinelis I will accept a counterexample to either case. | |
| Aug 15, 2023 at 13:35 | comment | added | Iosif Pinelis | What if this is disproved in one of the two cases? | |
| Aug 15, 2023 at 12:00 | history | edited | japalmer | CC BY-SA 4.0 |
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| Aug 15, 2023 at 3:23 | history | edited | japalmer | CC BY-SA 4.0 |
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| Aug 15, 2023 at 2:13 | comment | added | japalmer | @IosifPinelis I edited to specify that I am only asking for one of them. I'd prefer to keep all the information in the question since it may help with the solution. | |
| Aug 15, 2023 at 2:12 | history | edited | japalmer | CC BY-SA 4.0 |
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| Aug 14, 2023 at 20:19 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Aug 14, 2023 at 17:53 | comment | added | Iosif Pinelis | As in your previous post, there are two questions in this post as well. These two questions may be related but, looking at the previous post, they may require different treatments. According to MathOverflow guidelines, users should avoid answering multiple questions in one post. So, can you edit your post so as to retain just one question there? | |
| Aug 14, 2023 at 15:53 | comment | added | japalmer | Actually for the "stronger" functions the Stieltjes measure is $g''(x^2)$ not $f''(x)$. | |
| Aug 14, 2023 at 15:50 | history | edited | japalmer | CC BY-SA 4.0 |
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| Aug 14, 2023 at 15:07 | comment | added | japalmer | Being able to calculate the function of $r$ however, and show that the resulting function also satisfies a scale MLRP, allows the proof that the overall expected function, which is a derivative, still has only one zero, and thus the function has one maximum. The $r$ function was the subject of the other question you answered. | |
| Aug 14, 2023 at 15:04 | comment | added | japalmer | The equal area for all $\theta$ doesn't actually matter as any normalizing factor becomes a constant in the likelihood ratio. It's just interesting and unexpected. | |
| Aug 14, 2023 at 15:01 | comment | added | japalmer | I think that showing the MLRP for these piecewise kernels over the common non-zero support will show that they satisfy the MLRP, and that then the sum also satisfies the MLRP. | |
| Aug 14, 2023 at 14:58 | comment | added | japalmer | The stronger functions here can be written as scale mixtures of $1-(1-x^2)^2_+$ or $(x^2-1)^2_+$ with Stieltjes measure $f''(x)$. | |
| Aug 14, 2023 at 14:52 | comment | added | japalmer | I mean he integral over $\phi$ from $0$ to $\pi/2$ is independent of $\theta$, just a constant function of $f$ and $r$. This can be shown by writing these functions as scale mixtures of $1-(1-x^2)_+$ or $(x^2-1)_+$ with Stieltjes measure $-f'(x)/x$ or $f'(x)/x$. So the densities can be written as a sum of piecewise kernels, which depend on the $\theta$ and $r$, but somehow each $\theta$ and $r$ variant integrates to the same function of $r$ independent of $\theta$. | |
| Aug 14, 2023 at 14:43 | comment | added | Iosif Pinelis | What do you mean by"these densities indeed have the same measure for all $\theta$ whenever $g$ is concave or convex", and how to prove that? | |
| Aug 14, 2023 at 6:14 | history | edited | japalmer | CC BY-SA 4.0 |
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| Aug 14, 2023 at 5:32 | history | edited | japalmer | CC BY-SA 4.0 |
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| Aug 14, 2023 at 5:24 | history | edited | japalmer | CC BY-SA 4.0 |
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| Aug 14, 2023 at 5:19 | history | edited | japalmer | CC BY-SA 4.0 |
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| Aug 14, 2023 at 5:12 | history | edited | japalmer | CC BY-SA 4.0 |
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| Aug 14, 2023 at 5:06 | history | edited | japalmer | CC BY-SA 4.0 |
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| Aug 14, 2023 at 4:36 | history | edited | japalmer | CC BY-SA 4.0 |
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| Aug 14, 2023 at 4:30 | history | edited | japalmer | CC BY-SA 4.0 |
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| Aug 14, 2023 at 3:34 | history | edited | japalmer | CC BY-SA 4.0 |
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| Aug 14, 2023 at 3:15 | history | edited | japalmer | CC BY-SA 4.0 |
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| Aug 14, 2023 at 3:09 | history | edited | japalmer | CC BY-SA 4.0 |
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| Aug 14, 2023 at 2:33 | history | asked | japalmer | CC BY-SA 4.0 |