Timeline for Central limit theorem and the Pareto distribution
Current License: CC BY-SA 4.0
11 events
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| Sep 7, 2023 at 11:37 | comment | added | kjetil b halvorsen♦ | @SextusEmpiricus: Thanks, I will add to the answer | |
| Sep 7, 2023 at 7:00 | comment | added | Sextus Empiricus | So what I meant is: “The histogram in your figure still very much resembles a normal distribution”. It's just that it has long tales which mess up the scale (and these tails are not always sampled as your emperical sd's show) and that is why it doesn't match with the CLT approximation. (The problem is not the error in the SD estimates. Your observed sd estimate is so low because sampling from the heavy tails is very unlikely, but if you would have a sample that happens to match the SD=1, e.g. large R, then you would still not get a good match between the approximated curve and the histogram) | |
| Sep 7, 2023 at 6:23 | comment | added | Sextus Empiricus | Whether you use the theoretical values or the estimated values is not relevant, the point is that using the standard deviation (no matter what way) is problematic. If you use the CLT with a different scaling* then the CLT approximation is much better and the error mostly occurs for a fraction of a percent of the tails. The shape of the approximation is good for the central portion, but it is the scale that doesn't match. $$^*\text{for example:}\quad \frac{\bar{X} - \mu}{a_n} \\ \text{with $a_n$ something different than $\sqrt{n} \sigma$}$$ | |
| Sep 7, 2023 at 0:00 | comment | added | kjetil b halvorsen♦ | @SextusEmpiricus: Not so, the standardization in the simulation uses the theoretical mean and standard deviation, not estimates! | |
| Jul 25, 2023 at 23:08 | history | edited | kjetil b halvorsen♦ | CC BY-SA 4.0 |
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| Nov 21, 2021 at 13:33 | comment | added | Sextus Empiricus | In the example, the convergence is mostly bad/slow due to the standard deviation being so large due to some large values of which there are only a few (these large values in the tails say little about the shape of the density in the middle). When you scale the distribution based on something like the interquartile range then the convergence is faster. | |
| Sep 24, 2021 at 19:27 | history | edited | kjetil b halvorsen♦ | CC BY-SA 4.0 |
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| Oct 31, 2018 at 10:46 | history | edited | kjetil b halvorsen♦ | CC BY-SA 4.0 |
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| Apr 13, 2017 at 12:44 | history | edited | CommunityBot |
replaced http://stats.stackexchange.com/ with https://stats.stackexchange.com/
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| Jan 27, 2016 at 12:58 | history | edited | kjetil b halvorsen♦ | CC BY-SA 3.0 |
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| Jan 27, 2016 at 12:49 | history | answered | kjetil b halvorsen♦ | CC BY-SA 3.0 |