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Suppose one has $n$ real random variables $X_1, X_2, \dots, X_n$ from a certain distribution. Sort these random variables to get a sequence $Y_1, Y_2, \dots, Y_n$. What is known about the distribution, mean, variance, higher moments of the random variables $Y_i$? To be more specific:

1) Is it true that there is some sort of smoothing effect? As $i$ gets large the rv $Y_i$ has lower variance, say depending inversely on some increasing function of $i$?

2) It seems related to dependence assumptions. Can something more specific be said under assumptions of complete independence or under assumptions of negative dependence?

3) What general techniques exist, if any, to analyse the $Y_i$ in specific cases?

4) Suppose we look at this problem in a geometric setting. We are given $n$ points within the unit hypercube and the rv $X_i$ is the distance from point $i$ to a point chosen uar in the hypercube. Is something interesting known about the $Y_i$ in this case?

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  • $\begingroup$ There is a great deal known about the ranks when $\{X_i\}$ is an i.i.d. sample. You can find some information in Chapter 13 of van der Vaart's Asymptotic Statistics: books.google.com/… $\endgroup$
    – passerby51
    Commented May 14, 2012 at 20:02
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    $\begingroup$ If you assume independence, these are called order statistics. mathworld.wolfram.com/OrderStatistic.html I have no idea what you would hope to say if you don't assume independence. For any $i$, it is possible that $Y_i$ has the greatest variance among all of the order statistics, e.g., consider $X$ a Bernoulli random variable chosen so that $P(Y_i = 1) = 1/2$. $\endgroup$
    – Douglas Zare
    Commented May 14, 2012 at 20:12

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The following papers estimate the variances of order statistics:

Yang, H. (1982) "On the variances of median and some other order statistics." Bull. Inst. Math. Acad. Sinica, 10(2) pp. 197-204

Papadatos, N. (1995) "Maximum variance of order statistics." Ann. Inst. Statist. Math., 47(1) pp. 185-193

In particular, the variance of the median can't be greater than the variance of the population, although any other order statistic can have greater variance for Bernoulli random variables.

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The key word is "order statistics".

One thing that is known is that for any random variables $X_1,...,X_n$, if $Y_1,...,Y_n$ are the corresponding order statistics, then $\sum_i \text{Var}(Y_i) \le \sum_i \text{Var}(X_i)$.

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