evNormOrdStats {EnvStats} | R Documentation |
Compute the expected value of order statistics from a random sample from a standard normal distribution.
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
.
# 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