Timeline for concentration inequality for entropy from sample
Current License: CC BY-SA 3.0
10 events
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| Jan 21, 2016 at 23:40 | comment | added | Iosif Pinelis | Unfortunately, I don't know good answers to the questions in your latest comment. My suggestion at this point would be to play numerically with various $\mu$'s, and then perhaps you could come up with a plausible conjecture. | |
| Jan 21, 2016 at 23:06 | vote | accept | komark | ||
| Jan 21, 2016 at 23:02 | comment | added | komark | Thanks again for the detailed answer and especially for the Chernoff reference. I agree that the "probabilistic" part of the question is closed. However, in this form, (4) and (5) are difficult to apply. For instance, one could be interested in the question: for general $\mu$, for which $t$ we have $a_{+}(t) > \frac{1}{2}$? Is there a simpler (perhaps not quite tight) form for this $t$? Or alternatively, suppose we set $t = (1+\epsilon) H(\mu)$. Is there a simpler form for $a_{+}(t)$? | |
| Jan 21, 2016 at 16:38 | comment | added | Iosif Pinelis | If the deviations are only moderately large, that is, $|t|<<n^{-1/3}$, then, by the central limit theorem and Cramer's theorem for large deviations, the tail probabilities $P(S_n\ge|t|)$ and $P(S_n\le-|t|)$ are asymptotic to the corresponding normal ones; so, that normal asymptotic depends only on $n$, $t$, $EY$, and $EY^2$. | |
| Jan 21, 2016 at 16:26 | comment | added | Iosif Pinelis | If by "right" you mean "accurate and general enough", then the simplest parameter that controls the large deviations of $S_n$ (say, with $t$ fixed) is the function $m=m_\mu$ defined in $(3)$. You cannot make it simpler: I have now added the corresponding reference to Chernoff. | |
| Jan 21, 2016 at 16:11 | history | edited | Iosif Pinelis | CC BY-SA 3.0 |
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| Jan 21, 2016 at 5:18 | comment | added | komark | Thanks! But can we say something about the roots of equation (3) and (2) as function of $\mu$? I mean something simpler than that $h$ is determined by the roots of equation that is determined by $\mu$. I'm interested in: "what is the right and simple parameter that controls the deviations of S_n". | |
| Jan 21, 2016 at 5:13 | history | edited | Iosif Pinelis | CC BY-SA 3.0 |
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| Jan 21, 2016 at 4:51 | history | edited | Iosif Pinelis | CC BY-SA 3.0 |
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| Jan 21, 2016 at 4:35 | history | answered | Iosif Pinelis | CC BY-SA 3.0 |