Estimate Quantiles of a Generalized Extreme Value Distribution

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Description

Estimate quantiles of a generalized extreme value distribution.

Usage

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  eqgevd(x, p = 0.5, method = "mle", pwme.method = "unbiased", 
    tsoe.method = "med", 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 a generalized extreme value distribution (e.g., egevd). 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, scale, and threshold parameters. Possible values are "mle" (maximum likelihood; the default), "pwme" (probability-weighted moments), and "tsoe" (two-stage order-statistics estimator of Castillo and Hadi (1994)). See the DETAILS section of the help file for egevd 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 egevd for more information. This argument is ignored if method is not equal to "pwme".

tsoe.method

character string specifying the robust function to apply in the second stage of the two-stage order-statistics estimator when method="tsoe". Possible values are "med" (median; the default), and "lms" (least median of squares). See the DETAILS section of the help file for egevd for more information on these estimation methods. This argument is ignored if method is not equal to "tsoe".

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 egevd for more information. This argument is used only if method="tsoe", or if both method="pwme" and
pwme.method="plotting.position".

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 eqgevd returns estimated quantiles as well as estimates of the location, scale and threshold parameters.

Quantiles are estimated by 1) estimating the location, scale, and threshold parameters by calling egevd, and then 2) calling the function qgevd and using the estimated values for location, scale, and threshold.

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

Two-parameter extreme value distributions (EVD) have been applied extensively since the 1930's to several fields of study, including the distributions of hydrological and meteorological variables, human lifetimes, and strength of materials. The three-parameter generalized extreme value distribution (GEVD) was introduced by Jenkinson (1955) to model annual maximum and minimum values of meteorological events. Since then, it has been used extensively in the hydological and meteorological fields.

The three families of EVDs are all special kinds of GEVDs. When the shape parameter κ=0, the GEVD reduces to the Type I extreme value (Gumbel) distribution. (The function zTestGevdShape allows you to test the null hypothesis H_0: κ=0.) When κ > 0, the GEVD is the same as the Type II extreme value distribution, and when κ < 0 it is the same as the Type III extreme value distribution.

Hosking et al. (1985) compare the asymptotic and small-sample statistical properties of the PWME with the MLE and Jenkinson's (1969) method of sextiles. Castillo and Hadi (1994) compare the small-sample statistical properties of the MLE, PWME, and TSOE. Hosking and Wallis (1995) compare the small-sample properties of unbaised L-moment estimators vs. plotting-position L-moment estimators. (PWMEs can be written as linear combinations of L-moments and thus have equivalent statistical properties.) Hosking and Wallis (1995) conclude that unbiased estimators should be used for almost all applications.

Author(s)

Steven P. Millard (EnvStats@ProbStatInfo.com)

References

Castillo, E., and A. Hadi. (1994). Parameter and Quantile Estimation for the Generalized Extreme-Value Distribution. Environmetrics 5, 417–432.

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. (1984). Testing Whether the Shape Parameter is Zero in the Generalized Extreme-Value Distribution. Biometrika 71(2), 367–374.

Hosking, J.R.M. (1985). Algorithm AS 215: Maximum-Likelihood Estimation of the Parameters of the Generalized Extreme-Value Distribution. Applied Statistics 34(3), 301–310.

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.

Jenkinson, A.F. (1969). Statistics of Extremes. Technical Note 98, World Meteorological Office, Geneva.

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.

Macleod, A.J. (1989). Remark AS R76: A Remark on Algorithm AS 215: Maximum Likelihood Estimation of the Parameters of the Generalized Extreme-Value Distribution. Applied Statistics 38(1), 198–199.

Prescott, P., and A.T. Walden. (1980). Maximum Likelihood Estimation of the Parameters of the Generalized Extreme-Value Distribution. Biometrika 67(3), 723–724.

Prescott, P., and A.T. Walden. (1983). Maximum Likelihood Estimation of the Three-Parameter Generalized Extreme-Value Distribution from Censored Samples. Journal of Statistical Computing and Simulation 16, 241–250.

See Also

egevd, Generalized Extreme Value Distribution, Extreme Value Distribution, eevd,
estimate.object.

Examples

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  # Generate 20 observations from a generalized extreme value distribution 
  # with parameters location=2, scale=1, and shape=0.2, then compute the 
  # MLEs of location, shape,and threshold, and estimate the 90th percentile. 
  # (Note: the call to set.seed simply allows you to reproduce this example.)

  set.seed(498) 
  dat <- rgevd(20, location = 2, scale = 1, shape = 0.2) 
  eqgevd(dat, p = 0.9)

  #Results of Distribution Parameter Estimation
  #--------------------------------------------
  #
  #Assumed Distribution:            Generalized Extreme Value
  #
  #Estimated Parameter(s):          location = 1.6144631
  #                                 scale    = 0.9867007
  #                                 shape    = 0.2632493
  #
  #Estimation Method:               mle
  #
  #Estimated Quantile(s):           90'th %ile = 3.289912
  #
  #Quantile Estimation Method:      Quantile(s) Based on
  #                                 mle Estimators
  #
  #Data:                            dat
  #
  #Sample Size:                     20

  #----------

  # Clean up
  #---------
  rm(dat)

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