This question has come up at least twice now when I was trying to estimate something*. I could always write out the integral or find it computationally but I'm hoping someone will give me an exact answer, good approximation or at least decent upper bound.
Suppose I choose $k$ values, $X_1, \dots, X_k$ from $\mathcal{N}(0,\sigma^2)$. Let $L_1 = \text{min}\{ X_i \}$ and $L_{i+1} = \text{min} \{X_i | X_i > L_i \}$ so $L_1$ is the least value chosen, $L_2$ next least and so on.
So,
- What is the distribution of $L_2 - L_1$ exactly or approximately. It would be awesome if this was $|\mathcal{N}(0,\sigma^2) - \mathcal{N}(0,\sigma^2)|$ but I doubt it.
- Failing that what is a decent approximation of $E[L_2 - L_1]$
- Extra credit for a decent approximation to $L_1$ but I vaguely remember trying to figure this out once and discovering it was really nasty.
Just in case anyone is curious things like this come up in problems where you do something like pick an exam out of the stack and add 5 points to it and wish to approximate the expected change to the least score. If you know an easier way to approximate this I'm all ears.
To clarify I know how to set up the integrals what I'm looking for is an approximation that one can use without resorting to computational integration, just standard tables and calculator.