Order Statistic

Question

I have had hard time figuring out how to convert the uniform order statistic pdf into a beta distribution like how alpha and beta are determined.

 

If you answer this question, that would be greatly appreciated!

Answer 1

Hey Amon!

To determine the density of an order statistic, recall that the distribution for the k^th order statistic is:

f_X(k)(x) = n!/(k-1)!(n-k!) * [F(x)]^k-1[1-F(x)]^n-kf(x)

where f(.) is the density of X_(k).

In the case of a Uniform(0,1) random variable, the density is just 1, and the CDF is 0 for x<0, x for 0≤x≤1, and 1 for x>1.

Plugging things in, you get that following density for x between 0 and 1:

f_X(k)(x) = n!/(k-1)!(n-k!) * x^k-1[1-x]^n-k(1)

We can rewrite this so it more closely resembles the Beta(α,β) distribution:

f_X(k)(x) = Γ(k + (n-k+1)) / Γ(k)Γ(n-k+1)* x^k-1[1-x]^(n-k+1)-1

This makes it a little easier to see that α = k and β = n-k+1.