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This question already has an answer here:

I'm interested in finding the expected value for the kth ordered observation of a normally distributed variable with known standard deviation, mean and n. Could someone let me know the formula for that?

Thanks for your help

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marked as duplicate by Glen_b, whuber Mar 12 '15 at 0:38

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • $\begingroup$ Note that "assuming the order holds" is the same as saying "assuming the rank correlation is 1". But with that assumption, the first set of ranks becomes irrelevant to the question; you can drop that and instead deal with the simpler question about expected order statistics from a normal with known parameters. Please edit (since it will make your question clearer); when you do please also remove your name -- you already have a signature next to your gravatar on the right (posts are 'signed' for you already). $\endgroup$ – Glen_b Mar 11 '15 at 23:07
  • $\begingroup$ thanks for the tip on making the description a bit clearer Glen $\endgroup$ – Matt Mar 11 '15 at 23:22
  • $\begingroup$ Now the first link under "Related" in the sidebar to the right -> is directly relevant. $\endgroup$ – Glen_b Mar 12 '15 at 0:01
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For a sample of size $n$ from an absolutely continuous distribution, the general formula for the expected value of the $k$-th order statistic is

$$E[X_{k\,:\,n}] = \frac {n!}{(k-1)!(n-k)!}\int_{-\infty}^{\infty}x[F(x)]^{k-1}\cdot [1-F(x)]^{n-k}f(x){\rm d}x$$

where $F$ is the cumulative distribution function and $f$ is the probability density function. This is equivalent to

$$E[X_{k\,:\,n}] = E[F^{-1}(U_{k\,:\,n})]$$

where $U$ is a Uniform $U(0,1)$ random variable, and the "minus one" denotes the inverse, not the reciprocal.

This second expression uses the probability integral transform so one can compute the expected value of interest through simulation:

1) Generate a sample of size $n$ from a Uniform $U(0,1)$.
2) Pick the $k$-th order statistic of the sample.
3) Compute $F^{-1}(U_{k\,:\,n})$ and store, where $F$ is the CDF of interest.
4) Repeat steps 1-3 many times.
5) Take the average of the stored values.

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