eqevd {EnvStats}  R Documentation 
Estimate Quantiles of an Extreme Value (Gumbel) Distribution
Description
Estimate quantiles of an extreme value distribution.
Usage
eqevd(x, p = 0.5, method = "mle", pwme.method = "unbiased",
plot.pos.cons = c(a = 0.35, b = 0), digits = 0)
Arguments
x 
a numeric vector of observations, or an object resulting from a call to an
estimating function that assumes an extreme value distribution
(e.g., 
p 
numeric vector of probabilities for which quantiles will be estimated.
All values of 
method 
character string specifying the method to use to estimate the location and scale
parameters. Possible values are

pwme.method 
character string specifying what method to use to compute the
probabilityweighted moments when 
plot.pos.cons 
numeric vector of length 2 specifying the constants used in the formula for the
plotting positions when 
digits 
an integer indicating the number of decimal places to round to when printing out
the value of 
Details
The function eqevd
returns estimated quantiles as well as
estimates of the location and scale parameters.
Quantiles are estimated by 1) estimating the location and scale parameters by
calling eevd
, and then 2) calling the function
qevd
and using the estimated values for
location and scale.
Value
If x
is a numeric vector, eqevd
returns a
list of class "estimate"
containing the estimated quantile(s) and other
information. See estimate.object
for details.
If x
is the result of calling an estimation function, eqevd
returns a list whose class is the same as x
. The list
contains the same components as x
, as well as components called
quantiles
and quantile.method
.
Note
There are three families of extreme value distributions. The one
described here is the Type I, also called the Gumbel extreme value
distribution or simply Gumbel distribution. The name
“extreme value” comes from the fact that this distribution is
the limiting distribution (as n
approaches infinity) of the
greatest value among n
independent random variables each
having the same continuous distribution.
The Gumbel extreme value distribution is related to the
exponential distribution as follows.
Let Y
be an exponential random variable
with parameter rate=
\lambda
. Then X = \eta  log(Y)
has an extreme value distribution with parameters
location=
\eta
and scale=
1/\lambda
.
The distribution described above and assumed by eevd
is the
largest extreme value distribution. The smallest extreme value
distribution is the limiting distribution (as n
approaches infinity)
of the smallest value among
n
independent random variables each having the same continuous distribution.
If X
has a largest extreme value distribution with parameters
location=
\eta
and scale=
\theta
, then
Y = X
has a smallest extreme value distribution with parameters
location=
\eta
and scale=
\theta
. The smallest
extreme value distribution is related to the Weibull distribution
as follows. Let Y
be a Weibull random variable with
parameters
shape=
\beta
and scale=
\alpha
. Then X = log(Y)
has a smallest extreme value distribution with parameters location=
log(\alpha)
and scale=
1/\beta
.
The extreme value distribution has been used extensively to model the distribution of streamflow, flooding, rainfall, temperature, wind speed, and other meteorological variables, as well as material strength and life data.
Author(s)
Steven P. Millard (EnvStats@ProbStatInfo.com)
References
Castillo, E. (1988). Extreme Value Theory in Engineering. Academic Press, New York, pp.184–198.
Downton, F. (1966). Linear Estimates of Parameters in the Extreme Value Distribution. Technometrics 8(1), 3–17.
Forbes, C., M. Evans, N. Hastings, and B. Peacock. (2011). Statistical Distributions. Fourth Edition. John Wiley and Sons, Hoboken, NJ.
Greenwood, J.A., J.M. Landwehr, N.C. Matalas, and J.R. Wallis. (1979). Probability Weighted Moments: Definition and Relation to Parameters of Several Distributions Expressible in Inverse Form. Water Resources Research 15(5), 1049–1054.
Hosking, J.R.M., J.R. Wallis, and E.F. Wood. (1985). Estimation of the Generalized ExtremeValue Distribution by the Method of ProbabilityWeighted Moments. Technometrics 27(3), 251–261.
Johnson, N. L., S. Kotz, and N. Balakrishnan. (1995). Continuous Univariate Distributions, Volume 2. Second Edition. John Wiley and Sons, New York.
Landwehr, J.M., N.C. Matalas, and J.R. Wallis. (1979). Probability Weighted Moments Compared With Some Traditional Techniques in Estimating Gumbel Parameters and Quantiles. Water Resources Research 15(5), 1055–1064.
Tiago de Oliveira, J. (1963). Decision Results for the Parameters of the Extreme Value (Gumbel) Distribution Based on the Mean and Standard Deviation. Trabajos de Estadistica 14, 61–81.
See Also
eevd
, Extreme Value Distribution,
estimate.object
.
Examples
# Generate 20 observations from an extreme value distribution with
# parameters location=2 and scale=1, then estimate the parameters
# and estimate the 90'th percentile.
# (Note: the call to set.seed simply allows you to reproduce this example.)
set.seed(250)
dat < revd(20, location = 2)
eqevd(dat, p = 0.9)
#Results of Distribution Parameter Estimation
#
#
#Assumed Distribution: Extreme Value
#
#Estimated Parameter(s): location = 1.9684093
# scale = 0.7481955
#
#Estimation Method: mle
#
#Estimated Quantile(s): 90'th %ile = 3.652124
#
#Quantile Estimation Method: Quantile(s) Based on
# mle Estimators
#
#Data: dat
#
#Sample Size: 20
#
# Clean up
#
rm(dat)