Are there well known formulas for the order statistics of certain random distributions? Particularly the first and last order statistics of a normal
random variable, but a more general answer would also be appreciated.
Edit: To clarify, I am looking for approximating formulas that can be more-or-less explicitly evaluated, not the exact integral expression.
For example, I have seen the following two approximations for the first order statistic (ie the minimum) of a normal rv:
$e_{1:n} \geq \mu – \frac{n-1}{\sqrt{2n-1}}\sigma$
and
$e_{1:n} \approx \mu + \Phi^{-1} \left( \frac{1}{n+1} \right)\sigma$
The first of these, for $n=200$, gives approximately $e_{1:200} \geq \mu – 10\sigma$ which seems like a wildly loose bound.
The second gives $e_{1:200} \approx \mu – 2.58\sigma$ whereas a quick Monte Carlo gives $e_{1:200} \approx \mu – 2.75\sigma$, so it's not a bad approximation but not great either, and more importantly I don't have any intuition about where it comes from.
Any help?
Best Answer
The classic reference is Royston (1982)[1] which has algorithms going beyond explicit formulas. It also quotes a well-known formula by Blom (1958): $E(r:n) \approx \mu + \Phi^{-1}(\frac{r-\alpha}{n-2\alpha+1})\sigma$ with $\alpha=0.375$. This formula gives a multiplier of -2.73 for $n=200, r=1$.
[1]: Algorithm AS 177: Expected Normal Order Statistics (Exact and Approximate) J. P. Royston. Journal of the Royal Statistical Society. Series C (Applied Statistics) Vol. 31, No. 2 (1982), pp. 161-165