(Sandipan Dey, 26 December 2016)
 In this article, the concepts of the Order Statistics along with its applications will be discussed.
Order Statistics
 Definition: Consider a collection of i.i.d. continuous random variables X1,X2,…,Xn. If X(j) is the jth smallest of X1,X2,…,Xn, it’s called the jth order statistics. The 1st and the nth order statistics are the minimum and maximum of the variables, respectively. We are interested in the marginal density and expected value of the jth order statistics. The below figure shows the density of different order statistics in general and also for specific distributions (exponential and uniform).
 The following figure and animation show the density of the order statistics for n i.i.d random variables Xi∼U(0,1), for different values of n, along with the expected values of the order statistics as dotted vertical lines, quantities that we also shall be interested in.
 The following animation shows the density of the order statistics for n i.i.d random variables Xi∼Exp(5).

Application
 Now, let’s apply the order statistics concepts in the following settings of an auction.
 Let there be N potential buyers of some good.
 Their valuations are i.i.d. with U(0,1).
 The seller can offer the good
 at no cost
 at a posted price
 or can auction it off.
 The seller knows the distribution of valuations, but does not know the individual realizations.
 As shown in the following figure, the expected profit at the posted price depends on the CDF of the Nth order statistics of the valuations. The seller wants to maximize his expected profit and the optimal posted price is (1/(N+1))^(1/N), with the optimal expected profit as (N/(N+1))(1/(N+1))^(1/N).
 Again. as shown in the next figure, the profit at the 2nd price auction depends on the distribution of the (N−1)th order statistics of the valuations and the expected profit is computed to be (N−1)/(N+1).
 The following animation shows the distribution of the (N−1)th order statistics for N valuations for different values of N. The vertical dotted line shows the expected value as before.
 As can be seen from the following figure, as there are more and more potential buyers (N >= 3), the 2nd price auction becomes
more profitable in expectation than the optimal posted price.
 Now, let’s apply the order statistics concepts in the following settings of an auction.
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