Difference of two largest values from a set of i.i.d. random variables

Question

I have a set of i.i.d. random variables $(E_1, \ldots, E_N)$ that are exponentially distributed with parameter $\lambda$.

I guess, if $E_1$ lives in $(S,\mathcal{S},\mathbb{P})$, then the whole set lives in $(S^N,\bigotimes_{j=1}^N\mathcal{S},\mathbb{P}^N)$. I define $\Omega = S^N$.

Now, for a realization $\omega \in \Omega$, let $k_1(\omega)$ be the index such that $E_{k_1} (\omega)$ is the largest value from $(E_1(\omega), \ldots, E_N(\omega))$, and $k_2(\omega)$ the index such that $E_{k_2} (\omega)$ is the second largest value.

My question is: What is the distribution/density of $E_{k_1} - E_{k_2}$?

Well, I read in Gumbel's Statistics of extremes that if $\rho (x) = \lambda e^{-\lambda x}$ is the density of the $E$'s, then $E_{k_1}$ itself has a density

$N\rho (x) F(x)^{N-1}$

with $F$ being the cumulative function, i.e. $F(x) = \int_0^x \rho(y) dy$. This seems plausible to me.

The density for $E_{k_2}$ itself is

$N(N-1)\rho (x) (1- F(x)) F(x)^{N-2}$.

Usually, I determine the densities of sums and differences of independent random variables by Fourier or Laplace transform.

My problem is, that when I want to compute the density of $E_{k_1} - E_{k_2}$, then I have to take care about the dependecies of both values - and I do not know how.

Can anybody help?

Answer 1

Let $X \sim Exponential(\lambda)$ with pdf $f(x)$:

Let $X_n$ denote the largest order statistic, and $X_{n-1}$ the second largest.

Then, the joint pdf of $(X_{n-1}, X_n)$ is, say, $g(x_{n-1}, x_n)$:

where I am using the OrderStat function from the mathStatica package for Mathematica to automate the calculation.

We seek the cdf of $Y = X_n - X_{n-1}$, i.e. $P(Y<y)$, which is:

which is the cdf of an Exponential random variable with parameter $\lambda$. All done.

Notes

1. For a neat manual derivation, I suspect the memoryless property will show its hand.

2. As disclosure, I should add that I am one of the authors of the software used.