Timeline for Convergence rate of the central limit theorem near the center of the distribution
Current License: CC BY-SA 3.0
13 events
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| Feb 19, 2014 at 1:35 | comment | added | Daniel Soudry | Oops, I guess I should not rely on Wikipedia too much (just corrected the Edgeworth expansion wiki). Thanks for catching this! True, there is a correction term for lattice distributions (see paper "Edgeworth series for lattice distributions"). Unfortunately, this correction is always $n^{-1/2}$ on the lattice points. This is simply because the cumulative distribution (of the scaled distribution) have $O(n^{-1/2})$ jumps between lattice points. | |
| Feb 18, 2014 at 7:57 | comment | added | Brendan McKay | Careful, Davide's example has zero third moment and has convergence rate $n^{-1/2}$. Probably you are looking at an Edgeworth expansion that excludes lattice distributions. There is a different expansion for those. | |
| Feb 17, 2014 at 18:20 | comment | added | Daniel Soudry | Thanks again. For completeness, I will summarize my conclusions from the answer and all the helpful comments: 1) As a result of the Edgeworth expansion (en.wikipedia.org/wiki/Edgeworth_series): The convergence rate is $n^{-1/2}$, unless the third cumulant of the distribution is zero (as for the uniform distribution). In that case the convergence rate is $n^{-1}$, unless other cumulants are zero... and so on. 2) The reason that the approximation is bad at the tails is due to its relative precision - since the tails are usually are very small. | |
| Feb 17, 2014 at 3:32 | comment | added | Brendan McKay | ALso, if you can hold of P. Hall, Rates of convergence in the central limit theorem, there seems to be quite a lot of theory that is relevant. | |
| Feb 17, 2014 at 1:59 | comment | added | Daniel Soudry | Oh, never heard about these. Indeed they look very relevant. Thanks! | |
| Feb 16, 2014 at 8:14 | comment | added | Brendan McKay | Edgeworth expansions give extra terms that can be used to estimate the rate of convergence. | |
| S Feb 15, 2014 at 21:15 | history | suggested | Davide Giraudo |
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| Feb 15, 2014 at 21:14 | review | Suggested edits | |||
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| Feb 15, 2014 at 21:10 | vote | accept | Daniel Soudry | ||
| Feb 15, 2014 at 19:57 | history | edited | Daniel Soudry | CC BY-SA 3.0 |
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| Feb 15, 2014 at 19:53 | answer | added | Davide Giraudo | timeline score: 14 | |
| Feb 15, 2014 at 18:32 | history | edited | Daniel Soudry | CC BY-SA 3.0 |
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| Feb 15, 2014 at 18:10 | history | asked | Daniel Soudry | CC BY-SA 3.0 |