Timeline for Rate of convergence of a test statistic towards a Gaussian random variable
Current License: CC BY-SA 3.0
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| Apr 6, 2018 at 15:57 | history | edited | Przemek Repetowicz | CC BY-SA 3.0 |
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| Apr 6, 2018 at 9:33 | comment | added | Przemek Repetowicz | @enthdegree: Yes of course, the theorem in question does not describe this case and that is why I wanted to look into it to analyze it further.I am almost certain that there exist a whole family of functions $f()$ that lead to extremely slow convergence rates being related to iterated logarithms of $n$. | |
| Apr 5, 2018 at 18:38 | comment | added | Christian Chapman | I agree that if it does converge then the rate will be much slower than Berry-Esseen, but it seems that you've just confirmed the penultimate sentence in my first comment: With this normalization factor, the source distributions the statistic works over vary with the statistic's index $n$. The Berry-Esseen theorem doesn't make a statement about this situation. | |
| Apr 5, 2018 at 18:28 | comment | added | Przemek Repetowicz | Both in my example and in the other example linked to this question the second moments do not exist. This is compensated by different normalization factor,i.e $\sqrt{n \log(\log(n))}$ in my case and by $\sqrt{n \log(n)}$ in the case of the linked example. Now the really interesting question is the convergence ratio since it will turn out to be very slow, even much slower then in the linked example where it was $O(\log(\log(n)))/\log(n))$. Another lesson is that simulations are almost useless for exploring such cases since one has to go to very large $n$ values to see anything. | |
| Apr 5, 2018 at 18:18 | history | edited | Przemek Repetowicz | CC BY-SA 3.0 |
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| Apr 5, 2018 at 18:07 | comment | added | Christian Chapman | The Berry-Esseen theorem's statement involves random variables who have finite first three moments. But it seems like your random variable doesn't even have a variance... If you mean for the $\log\log$ term in the denominator to be a correction for this, then the distributions the statistic $S_n$ works on now vary with $n$. How is this going to lead to a counterexample? | |
| Apr 5, 2018 at 17:29 | history | edited | Przemek Repetowicz | CC BY-SA 3.0 |
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| Apr 5, 2018 at 9:53 | history | edited | Przemek Repetowicz | CC BY-SA 3.0 |
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| Apr 4, 2018 at 18:17 | history | edited | Przemek Repetowicz | CC BY-SA 3.0 |
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| Apr 4, 2018 at 16:23 | history | edited | Przemek Repetowicz | CC BY-SA 3.0 |
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| Apr 4, 2018 at 14:53 | review | First posts | |||
| Apr 4, 2018 at 14:56 | |||||
| Apr 4, 2018 at 14:52 | history | asked | Przemek Repetowicz | CC BY-SA 3.0 |