Let \(\underline{z} = z_1, z_2, \ldots, z_n\) denote a vector of \(n\)
observations from a normal distribution with parameters
mean=0 and sd=1. That is, \(\underline{z}\) denotes a vector of
\(n\) observations from a standard normal distribution. Let
\(z_{(r)}\) denote the \(r\)'th order statistic of \(\underline{z}\),
for \(r = 1, 2, \ldots, n\). The probability density function of
\(z_{(r)}\) is given by:
$$f_{r,n}(t) = \frac{n!}{(r-1)!(n-r)!} [\Phi(t)]^{r-1} [1 - \Phi(t)]^{n-r} \phi(t) \;\;\;\;\;\; (1)$$
where \(\Phi\) and \(\phi\) denote the cumulative distribution function and
probability density function of the standard normal distribution, respectively
(Johnson et al., 1994, p.93). Thus, the expected value of \(z_{(r)}\) is given by:
$$E(r, n) = E[z_{(r)}] = \int_{-\infty}^{\infty} t f_{r,n}(t) dt \;\;\;\;\;\; (2)$$
It can be shown that if \(n\) is odd, then
$$E[(n+1)/2, n] = 0 \;\;\;\;\;\; (3)$$
Also, for all values of \(n\),
$$E(r, n) = -E(n-r, n) \;\;\;\;\;\; (4)$$
The function evNormOrdStatsScalar computes the value of \(E(r,n)\) for
user-specified values of \(r\) and \(n\).
The function evNormOrdStats computes the values of \(E(r,n)\) for all
values of \(r\) for a user-specified value of \(n\).
For large values of \(n\), the function evNormOrdStats with
approximate=FALSE may take a long time to execute. When
approximate=TRUE, evNormOrdStats and evNormOrdStatsScalar
use the following approximation to \(E(r,n)\), which was proposed by
Blom (1958, pp. 68-75):
$$E(r, n) \approx \Phi^{-1}(\frac{r - 3/8}{n + 1/4}) \;\;\;\;\;\; (5)$$
This approximation is quite accurate. For example, for \(n \ge 2\), the
approximation is accurate to the first decimal place, and for \(n \ge 9\) it
is accurate to the second decimal place.