Rank $k$ of a sequence of random variables - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T01:54:37Z http://mathoverflow.net/feeds/question/96940 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/96940/rank-k-of-a-sequence-of-random-variables Rank $k$ of a sequence of random variables Nirman 2012-05-14T19:42:42Z 2012-05-14T20:39:45Z <p>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:</p> <p>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 $i$?</p> <p>2) It seems related to dependence assumptions. Can something more specific be said under assumptions of complete independence or under assumptions of negative dependence?</p> <p>3) What general techniques exist, if any, to analyse the $Y_i$ in specific cases?</p> <p>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? </p> http://mathoverflow.net/questions/96940/rank-k-of-a-sequence-of-random-variables/96943#96943 Answer by Robert Israel for Rank $k$ of a sequence of random variables Robert Israel 2012-05-14T20:19:39Z 2012-05-14T20:19:39Z <p>The key word is "order statistics".</p> <p>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 $\sum_i \text{Var}(Y_i) \le \sum_i \text{Var}(X_i)$. </p> http://mathoverflow.net/questions/96940/rank-k-of-a-sequence-of-random-variables/96945#96945 Answer by Douglas Zare for Rank $k$ of a sequence of random variables Douglas Zare 2012-05-14T20:39:45Z 2012-05-14T20:39:45Z <p>The following papers estimate the variances of order statistics:</p> <p>Yang, H. (1982) "On the variances of median and some other order statistics." Bull. Inst. Math. Acad. Sinica, 10(2) pp. 197-204 </p> <p>Papadatos, N. (1995) <a href="http://www.ism.ac.jp/editsec/aism/pdf/047_1_0185.pdf" rel="nofollow">"Maximum variance of order statistics."</a> Ann. Inst. Statist. Math., 47(1) pp. 185-193 </p> <p>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.</p>