In this article, the concepts of the Order Statistics along with its applications will be discussed. These concepts were discussed in a lecture video in the edX course MITx 14.310x Data Analysis for Social Scientists.

Order Statistics

Definition

  • Let’s consider a collection of i.i.d. contiuous random variables \(X_1,X_2,\ldots,X_n\). If \(X_{(j)}\) is the \(j^{th}\) smallest of \(X_1,X_2,\ldots,X_n\), it’s called the \(j^{th}\) order statistics.

  • The \(1^{st}\) 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 distributions).

  • The following figure and animation show the density of the order statistics for \(n\) i.i.d random variables \(X_i \sim U(0,1)\), for different values of \(n\), along with the expected values of the order statistics as dotted vertical lines, quantities that we shall also be interested in.

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 \(\left(\frac{1}{N+1}\right)^{\frac{1}{N}}\), with the optimal expected profit as \(\frac{N}{N+1}\left(\frac{1}{N+1}\right)^{\frac{1}{N}}\).

    • 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 \(\frac{N-1}{N+1}\).

  • The following figure shows the distribution of the \((N-1)^{th}\) order statistics for \(N\) valuations for different \(N\). The vertical dotted line shows the expected value as before.