(This question was previously posted to sci.math.research. I only received one reply; sadly the advice therein conflicted with section 9.1 of H.A. David's "Order Statistics" - and probably with the fact that there was such a field of study as "r-extreme order statistics" - hence my reposting it here.)(adsbygoogle = window.adsbygoogle || []).push({});

I've been studying some cryptographic research in which the asymptotic normal distribution of the empirical sample quartile of order q is used to construct statistical models of the amount of data required for a successful cryptanalysis.

The main issue I have is that, while I'm pretty sure that such models have continued to be used for order statistics X_i (with i near to n) where the asymptotic normal distribution is inaccurate and where something based on extreme-value theory for the mth extremes would have been better, I don't have any idea as to how to compute an estimate for the value of i (or indeed q) above which the asymptotic normal might be considered suspect.

As an example, I'm currently dealing with the situation X_1 ≤ X_2 ...≤ X_n, where n = 2^{41}-1 = 2,199,023,255,551. In particular, I'm trying to work out whether the asymptotic normal is likely to be adequate when drawing conclusions about the top 2^{17} = 131,072 values or not - and while this seems a high m for m-th-extreme, it's not so high in relation to n, and this would mean I was dealing with the top 0.000006388% of values.

Can anyone give me some advice here?

Many thanks,

James McLaughlin.

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# Extreme value theory and limiting distributions for i.i.d. order statistics

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