I would like to understand the relationship between the asymptotic moments of order statistics and the moments of the distribution of the mother distribution. I will appreciate any references on this matter.
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1Asymptotic for which limit? – Did Jul 02 '11 at 09:20
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as the number of samples goes to infinity – user12847 Jul 02 '11 at 09:39
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Then the answer very much depends on whether (what you call) the mother distribution is bounded or not, and on the rank of the order statistics you consider, either at the top, or at the bottom, or somewhere in the middle of the sample. For the latter, you might wish to consult the formula given here: http://en.wikipedia.org/wiki/Order_statistic#Large_sample_sizes – Did Jul 02 '11 at 09:47
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Yes, and what is the relation of this asymptotic variance and the variance of the mother distribution? – user12847 Jul 02 '11 at 10:17
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Bis repetita: the asymptotic variance of what? $X_{(1)}$? $X_{(n)}$? $X_{(k)}$ for $k=pn+o(n)$ and $p$ fixed in $(0,1)$? These all lead to very different answers. For the latter, the answer is on the WP page I linked to. – Did Jul 02 '11 at 11:00
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I am aware of the formula of the asymptotic variance given the underlying distribution. My question is, when the variance of the underlying distribution increases, how will it affect the asymptotic variance? – user12847 Jul 02 '11 at 16:28
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Ter repetita (and last): the asymptotic variance of what? – Did Jul 02 '11 at 16:50
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intermediate order statistics. – user12847 Jul 02 '11 at 16:50