eqevd {EnvStats}

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., eevd). If x is a numeric vector, missing (NA), undefined (NaN), and infinite (Inf, -Inf) values are allowed but will be removed.

p

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.

method

character string specifying the method to use to estimate the location and scale parameters. Possible values are "mle" (maximum likelihood; the default), "mme" (methods of moments), "mmue" (method of moments based on the unbiased estimator of variance), and "pwme" (probability-weighted moments). See the DETAILS section of the help file for eevd for more information on these estimation methods.

pwme.method

character string specifying what method to use to compute the probability-weighted moments when method="pwme". The possible values are "ubiased" (method based on the U-statistic; the default), or "plotting.position" (method based on the plotting position formula). See the DETAILS section of the help file for eevd for more information. This argument is ignored if method is not equal to "pwme".

plot.pos.cons

numeric vector of length 2 specifying the constants used in the formula for the plotting positions when method="pwme" and
pwme.method="plotting.position". The default value is
plot.pos.cons=c(a=0.35, b=0). If this vector has a names attribute with the value c("a","b") or c("b","a"), then the elements will be matched by name in the formula for computing the plotting positions. Otherwise, the first element is mapped to the name "a" and the second element to the name "b". See the DETAILS section of the help file for eevd for more information. This argument is ignored if method is not equal to "pwme" or if pwme.method="ubiased".

digits

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.

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 ([e-mail])

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 Extreme-Value Distribution by the Method of Probability-Weighted 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)