# Some Statistics Concepts: Order Statistics and Application in Auction

• 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 j-th smallest of X1,X2,,Xn, it’s called the j-th order statistics. The 1st and the n-th order statistics are the minimum and maximum of the variables, respectively. We are interested in the marginal density and expected value of the j-th 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 XiU(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 XiExp(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 N-th 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 (N1)-th order statistics of the valuations and the expected profit is computed to be (N1)/(N+1).
• The following animation shows the distribution of the (N1)-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.