eqweibull.RdEstimate quantiles of a Weibull distribution.
eqweibull(x, p = 0.5, method = "mle", digits = 0)a numeric vector of observations, or an object resulting from a call to an
estimating function that assumes a Weibull distribution
(e.g., eweibull). If x is a numeric vector,
missing (NA), undefined (NaN), and infinite (Inf, -Inf)
values are allowed but will be removed.
numeric vector of probabilities for which quantiles will be estimated.
All values of p must be between 0 and 1. The default value is p=0.5.
character string specifying the method of estimating the distribution parameters.
Possible values are
"mle" (maximum likelihood; the default), "mme" (methods of moments),
and "mmue" (method of moments based on the unbiased estimator of variance).
See the DETAILS section of the help file for eweibull for more
information.
an integer indicating the number of decimal places to round to when printing out
the value of 100*p. The default value is digits=0.
The function eqweibull returns estimated quantiles as well as
estimates of the shape and scale parameters.
Quantiles are estimated by 1) estimating the shape and scale parameters by
calling eweibull, and then 2) calling the function
qweibull and using the estimated values for
shape and scale.
If x is a numeric vector, eqweibull 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, eqweibull
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.
Forbes, C., M. Evans, N. Hastings, and B. Peacock. (2011). Statistical Distributions. Fourth Edition. John Wiley and Sons, Hoboken, NJ.
Johnson, N. L., S. Kotz, and N. Balakrishnan. (1994). Continuous Univariate Distributions, Volume 1. Second Edition. John Wiley and Sons, New York.
The Weibull distribution is named after the Swedish physicist Waloddi Weibull, who used this distribution to model breaking strengths of materials. The Weibull distribution has been extensively applied in the fields of reliability and quality control.
The exponential distribution is a special case of the
Weibull distribution: a Weibull random variable with parameters shape=\(1\)
and scale=\(\beta\) is equivalent to an exponential random variable with
parameter rate=\(1/\beta\).
The Weibull distribution is related to the
Type I extreme value (Gumbel) distribution as follows:
if \(X\) is a random variable from a Weibull distribution with parameters
shape=\(\alpha\) and scale=\(\beta\), then
$$Y = -log(X) \;\;\;\; (10)$$
is a random variable from an extreme value distribution with parameters
location=\(-log(\beta)\) and scale=\(1/\alpha\).
# Generate 20 observations from a Weibull distribution with parameters
# shape=2 and scale=3, then estimate the parameters via maximum likelihood,
# and estimate the 90'th percentile.
# (Note: the call to set.seed simply allows you to reproduce this example.)
set.seed(250)
dat <- rweibull(20, shape = 2, scale = 3)
eqweibull(dat, p = 0.9)
#>
#> Results of Distribution Parameter Estimation
#> --------------------------------------------
#>
#> Assumed Distribution: Weibull
#>
#> Estimated Parameter(s): shape = 2.673098
#> scale = 3.047762
#>
#> Estimation Method: mle
#>
#> Estimated Quantile(s): 90'th %ile = 4.163755
#>
#> Quantile Estimation Method: Quantile(s) Based on
#> mle Estimators
#>
#> Data: dat
#>
#> Sample Size: 20
#>
#Results of Distribution Parameter Estimation
#--------------------------------------------
#
#Assumed Distribution: Weibull
#
#Estimated Parameter(s): shape = 2.673098
# scale = 3.047762
#
#Estimation Method: mle
#
#Estimated Quantile(s): 90'th %ile = 4.163755
#
#Quantile Estimation Method: Quantile(s) Based on
# mle Estimators
#
#Data: dat
#
#Sample Size: 20
#----------
# Clean up
#---------
rm(dat)