Order Statistics of Exponential RVs June 5, 2007

Posted by Peter in Exam 1/P, Exam 3/MLC, Exam 4/C.

Here’s a question I read from the AoPS forum, and answered therein:

In analyzing the risk of a catastrophic event, an insurer uses the exponential distribution with mean $\theta$ as the distribution of the time until the event occurs. The insured had n independent catastrophe policies of this type. Find the expected time until the insured will have the first catastrophe claim.

The sum $S = X_{1}+X_{2}+\cdots+X_{n}$ of n independent and identically distributed exponential random variables $X_{1}, X_{2}, \ldots, X_{n}$ is gamma distributed. Specifically, if $X_{k}$ are exponential with mean $\theta$, then S is gamma with mean $n\theta$ and variance $n\theta^{2}$.

It’s noteworthy that the sum S is a random variable that describes the time until the n-th claim if claims followed a Poisson process (whose interarrival times are exponentially distributed).

However, according to the model you specified, the events are not interarrival times, but rather they run concurrently. So the time until the k-th event is NOT gamma; rather, it is the k-th order statistic $Y_{k}$. Fortunately, the first such order statistic $Y_{1}$ is exponential, which we show by recalling that $Y_{k}$ has PDF

$\displaystyle f_{Y_{k}}(x) = \frac{n!}{(k-1)!(n-k)!}F_{X}(x)^{k-1}(1-F_{X}(x))^{n-k}f_{X}(x).$

If k = 1, we immediately obtain

$\displaystyle f_{Y_{1}}(x) = n(e^{-x/\theta})^{n-1}\frac{1}{\theta}e^{-x/\theta} = \frac{n}{\theta}e^{-x(n/\theta)},$

which is exponential with mean $\theta/n$. Note that this answer makes sense because as the number of policies n held by the insured increases, the expected waiting time until the first claim decreases. This would not be the case if one and only one policy at a time were in force at all times until the last policy claim–such a scenario would correspond to the gamma distribution previously mentioned.

To check your understanding, here are some exercises:

1. What is the PDF of the second order statistic $Y_{2}$, and what does it represent?
2. Which of the $Y_{k}$ belong to the gamma or exponential family of distributions?
3. Prove that $\displaystyle {\rm E}[Y_{k}] = \theta \sum_{j=1}^{k}\frac{1}{n-k+j}.$