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For random variables $X_1, \cdots, X_n$, we denote the order statistics by

\begin{align} X_{(1)} & = \min (X_1,\ldots, X_n) \\[6pt] X_{(2)} & = \text{second-smallest of } X_1,\ldots, X_n \\ & \,\,\,\vdots \\ X_{(n)} & = \max (X_1,\ldots, X_n) \end{align}

Now let $Z_1,\ldots,Z_n$ be $n$ random samples drawn from a standard Normal distribution. Let $\mu_{i:n}$ be the expected value of the $i$-th order statistic. As far as I know, there is no closed form formula for $\mu_{i:n}$, and Blom (1958)[1] provides a simple but yet fairly accurate approximation for $\mu_{i:n}$: namely

$$\mu_{i:n}=\Phi^{-1} \left(\frac{i-\frac{3}{8}}{n+\frac{1}{4}} \right),$$ where $\Phi$ is the cdf of the standard Normal.

My question is, is there any advances in the approximation for $\mu_{i:n}$ since Blom or is there any database for the values of $\mu_{i:n}$?

Any help would be appreciated, thanks.

Reference:

[1] G. Blom. Statistical Estimates and Transformed Beta Variables. 1958.

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    $\begingroup$ You should be able to generate accurate tables using numerical integration from many software packages (Mathematica, Maple, MATLAB, etc.) Is there a particular range of values of $n$ that are of interest? $\endgroup$ Commented Jul 4, 2024 at 19:45
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    $\begingroup$ Apart from the extreme two order statistics ($(1)$ and $(n)$), Blom's approach works really well (though the value of $a$ in $\frac{i-a}{n+1-2a}$ needs to vary slowly with sample size, e.g. from about 0.37 at n=5, about 0.4 at n=45, to about 0.42 at n=250, and I believe it converges to 0.5 as n increases). There's still a little pattern but it's surprisingly close. For the expected values of the sample min and max there's a number of useful papers. Tippett's 1927 paper was quite good but there's been quite a few papers since. $\endgroup$ Commented Jul 5, 2024 at 5:16
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    $\begingroup$ Rather than saying "$n$ random samples", you should write "a random sample of size $n.$" $\endgroup$ Commented Jul 6, 2024 at 2:31
  • $\begingroup$ FWIW, I believe there are closed formulas, at least for some of the expectations of the order statistics, up through $n=5.$ The methods to obtain them are algebraic and recursive rather than analytic, as described in work of Balakrishnan. $\endgroup$ Commented Jul 6, 2024 at 15:08

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Here is an example of using numerical integration to find the expected values using Mathematica which is more accurate than Blom's equation. (Note: to make a "nice" table there are formatting commands included below that aren't necessary if you just want an individual expectation.)

t = ConstantArray[".", {9, 10}];
Do[Do[t[[n - 1, i]] = ToString[NumberForm[
  NIntegrate[x  PDF[OrderDistribution[{NormalDistribution[0, 1], n}, i], x], {x, -Infinity, Infinity}],
  {6, 5}]], {i, 1, n}], {n, 2, 10}]
TableForm[t, TableHeadings -> {"n = " <> ToString[#] & /@ Range[2, 10], 
   "i = " <> ToString[#] & /@ Range[10]}]

Table of expected values

The basic command could be written as a function:

expectation[n_, i_] := 
  NIntegrate[x PDF[OrderDistribution[{NormalDistribution[0, 1], n}, i], x],
   {x, -Infinity, Infinity}]

expectation[5, 4]
(* 0.495019 *)

Exact values of the mean are available for just a few combinations of $n$ and $i$. Here are expected minimums for n = 2, 3, 4 and 5.

Mean[OrderDistribution[{NormalDistribution[0, 1], 2}, 1]]

$$-\frac{1}{\sqrt{\pi }}$$

Mean[OrderDistribution[{NormalDistribution[0, 1], 3}, 1]]

$$-\frac{3}{2 \sqrt{\pi }}$$

Mean[OrderDistribution[{NormalDistribution[0, 1], 4}, 1]]

$$-\frac{6 \tan ^{-1}\left(\sqrt{2}\right)}{\pi ^{3/2}}$$

Mean[OrderDistribution[{NormalDistribution[0, 1], 5}, 1]]

$$\frac{5}{2 \sqrt{\pi }}-\frac{15 \tan ^{-1}\left(\sqrt{2}\right)}{\pi ^{3/2}}$$

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