Description Usage Arguments Details Value Note Author(s) References See Also Examples

View source: R/evNormOrdStats.R

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

1 2 3 | ```
evNormOrdStats(n = 1, approximate = FALSE)
evNormOrdStatsScalar(r = 1, n = 1, approximate = FALSE)
``` |

`n` |
positive integer indicating the sample size. |

`r` |
positive integer between |

`approximate` |
logical scalar indicating whether to use the Blom score approximation (Blom, 1958).
The default value is |

Let *\underline{z} = z_1, z_2, …, 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, …, n*. The probability density function of
*z_{(r)}* is given by:

*f_{r,n}(t) = \frac{n!}{(r-1)!(n-r)!} [Φ(t)]^{r-1} [1 - Φ(t)]^{n-r} φ(t) \;\;\;\;\;\; (1)*

where *Φ* and *φ* 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_{-∞}^{∞} 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 Φ^{-1}(\frac{r - 3/8}{n + 1/4}) \;\;\;\;\;\; (5)*

This approximation is quite accurate. For example, for *n ≥ 2*, the
approximation is accurate to the first decimal place, and for *n ≥ 9* it
is accurate to the second decimal place.

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.

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`

.

Steven P. Millard (EnvStats@ProbStatInfo.com)

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`

.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 | ```
# 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
``` |

EnvStats documentation built on May 20, 2017, 12:28 a.m.

Questions? Problems? Suggestions? Tweet to @rdrrHQ or email at ian@mutexlabs.com.

Please suggest features or report bugs in the GitHub issue tracker.

All documentation is copyright its authors; we didn't write any of that.

Embedding an R snippet on your website

Add the following code to your website.

For more information on customizing the embed code, read Embedding Snippets.