# evNormOrdStats

0th

Percentile

##### Expected Value of Order Statistics from Random Sample from Standard Normal Distribution

Compute the expected value of order statistics from a random sample from a standard normal distribution.

Keywords
distribution
##### Usage
evNormOrdStats(n = 1, approximate = FALSE)

evNormOrdStatsScalar(r = 1, n = 1, approximate = FALSE)
##### Arguments
n
positive integer indicating the sample size.
r
positive integer between 1 and n specifying the order statistic for which to compute the expected value.
approximate
logical scalar indicating whether to use the Blom score approximation (Blom, 1958). The default value is FALSE.
##### Details

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.

##### Value

• For evNormOrdStats: a numeric vector of length n containing the expected values of all the order statistics for a random sample of n standard normal deviates. For evNormOrdStatsScalar: a numeric scalar containing the expected value of the r'th order statistic from a random sample of n standard normal deviates.

##### Note

The expected values of normal order statistics are used to construct normal quantile-quantile (Q-Q) plots (see qqPlot) and to compute goodness-of-fit statistics (see gofTest). Usually, however, approximations are used instead of exact values. The functions evNormOrdStats and evNormOrdStatsScalar have been included mainly because evNormOrdStatsScalar is called by elnorm3 and predIntNparSimultaneousTestPower.

##### References

Johnson, N. L., S. Kotz, and N. Balakrishnan. (1994). Continuous Univariate Distributions, Volume 1. Second Edition. John Wiley and Sons, New York, pp. 93--99. Royston, J.P. (1982). Algorithm AS 177. Expected Normal Order Statistics (Exact and Approximate). Applied Statistics 31, 161--165.

Normal, elnorm3, predIntNparSimultaneousTestPower, gofTest, qqPlot.

##### Aliases
• evNormOrdStats
• evNormOrdStatsScalar
##### Examples
# Compute the expected value of the minimum for a random sample of size 10
# from a standard normal distribution:

evNormOrdStatsScalar(r = 1, n = 10)
#[1] -1.538753

#----------

# Compute the expected values of all of the order statistics for a random sample
# of size 10 from a standard normal distribution:

evNormOrdStats(10)
#[1] -1.5387527 -1.0013570 -0.6560591 -0.3757647 -0.1226888
#[6]  0.1226888  0.3757647  0.6560591  1.0013570  1.5387527

# Compare the above with Blom (1958) scores:

evNormOrdStats(10, approx = TRUE)
#[1] -1.5466353 -1.0004905 -0.6554235 -0.3754618 -0.1225808
#[6]  0.1225808  0.3754618  0.6554235  1.0004905  1.5466353
Documentation reproduced from package EnvStats, version 2.1.0, License: GPL (>= 3)

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