Timeline for Trying to bound one functional by another functional
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 27, 2021 at 4:12 | comment | added | Fei Cao | Thank you very much! | |
| Jun 27, 2021 at 2:47 | comment | added | Iosif Pinelis | @FeiCao : The assumption that the mean $\mu$ was fixed was not stated in your OP. It was only said that $\mu>0$. However, even assuming $\mu$ is fixed, Remark 1 in my answer gives essentially the same conclusion, that $f=0$. I have added details on this. | |
| Jun 27, 2021 at 2:44 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
added 896 characters in body
|
| Jun 26, 2021 at 2:39 | vote | accept | Fei Cao | ||
| Jun 25, 2021 at 23:24 | comment | added | Fei Cao | In fact, I thought your answer is still missing something. In my OP I indeed said that $\rho$ has some fixed mean $\mu > 0$, so your example will only work when $a = \mu$ (you are forbidden to take any real $a>0$). With your argument, we can only say that $f(\mu) = 0$, am I right? | |
| Jun 25, 2021 at 20:06 | comment | added | Fei Cao | I agree with you. Thank you! | |
| Jun 25, 2021 at 20:06 | vote | accept | Fei Cao | ||
| Jun 25, 2021 at 23:25 | |||||
| Jun 25, 2021 at 19:59 | comment | added | Iosif Pinelis | @FeiCao : As briefly noted in my answer, no degree of smoothness can possibly help here, because any distribution can be appropriately approximated by an arbitrarily smooth one. If you want to impose additional, "structural" conditions on the distribution (such as, say, the log-concavity of the pdf), that would be quite a different question, to be posted separately. For now, your question, as posted, has been fully answered. | |
| Jun 25, 2021 at 19:48 | comment | added | Fei Cao | Thanks for your answer! But I don't expect that the relation $H\geq f(G)$ to hold for all p.d.f. $\rho$ whose support is contained in $[0,\infty)$. For instance, is it possible to show something like this for "smooth" $\rho$ (instead of being a combination of Dirac masses)? | |
| Jun 25, 2021 at 19:34 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
added 213 characters in body
|
| Jun 25, 2021 at 19:27 | history | answered | Iosif Pinelis | CC BY-SA 4.0 |