Test Whether the Shape Parameter of a Generalized Extreme Value Distribution is Equal to 0

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Description

Estimate the shape parameter of a generalized extreme value distribution and test the null hypothesis that the true value is equal to 0.

Usage

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  zTestGevdShape(x, pwme.method = "unbiased", 
    plot.pos.cons = c(a = 0.35, b = 0), alternative = "two.sided")

Arguments

x

numeric vector of observations. Missing (NA), undefined (NaN), and infinite (Inf, -Inf) values are allowed but will be removed.

pwme.method

character string specifying the method of estimating the probability-weighted moments. Possible values are "unbiased" (method based on the U-statistic; the default), and "plotting.position" (plotting position). See the help file for egevd for more information on these estimation methods.

plot.pos.cons

numeric vector of length 2 specifying the constants used in the formula for the plotting positions. 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 help file for egevd for more information. This argument is ignored if pwme.method is not equal to
"plotting.position".

alternative

character string indicating the kind of alternative hypothesis. The possible values are "two.sided" (shape not equal to 0; the default), "less" (shape less than 0), and "greater" (shape greater than 0).

Details

Let \underline{x} = x_1, x_2, …, x_n be a vector of n observations from a generalized extreme value distribution with parameters location=η, scale=θ, and shape=κ. Furthermore, let \hat{κ}_{pwme} denote the probability-weighted moments estimator (PWME) of the shape parameter κ (see the help file for egevd). Then the statistic

z = \frac{\hat{κ}_{pwme}}{√{0.5633/n}} \;\;\;\;\;\; (1)

is asymptotically distributed as a N(0,1) random variable under the null hypothesis H_0: κ = 0 (Hosking et al., 1985). The function zTestGevdShape performs the usual one-sample z-test using the statistic computed in Equation (1). The PWME of κ may be computed using either U-statistic type probability-weighted moments estimators or plotting-position type estimators (see egevd). Although Hosking et al. (1985) base their statistic on plotting-position type estimators, Hosking and Wallis (1995) recommend using the U-statistic type estimators for almost all applications.

This test is only asymptotically correct. Hosking et al. (1985), however, found that the α-level is adequately maintained for samples as small as n = 25.

Value

A list of class "htest" containing the results of the hypothesis test. See the help file for htest.object for details.

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. 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) introduced the test used by the function zTestGevdShape to test the null hypothesis H_0: κ = 0. They found this test has power comparable to the modified likelihood-ratio test, which was found by Hosking (1984) to be the best overall test the thirteen tests he considered.

Fill and Stedinger (1995) denote this test the “kappa test” and compare it with the L-Cs test suggested by Chowdhury et al. (1991) and the probability plot correlation coefficient goodness-of-fit test for the Gumbel distribution given by Vogel (1986) (see the sub-section for test="ppcc" under the Details section of the help file for gofTest).

Author(s)

Steven P. Millard (EnvStats@ProbStatInfo.com)

References

Chowdhury, J.U., J.R. Stedinger, and L. H. Lu. (1991). Goodness-of-Fit Tests for Regional Generalized Extreme Value Flood Distributions. Water Resources Research 27(7), 1765–1776.

Fill, H.D., and J.R. Stedinger. (1995). L Moment and Probability Plot Correlation Coefficient Goodness-of-Fit Tests for the Gumbel Distribution and Impact of Autocorrelation. Water Resources Research 31(1), 225–229.

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., and J.R. Wallis (1995). A Comparison of Unbiased and Plotting-Position Estimators of L Moments. Water Resources Research 31(8), 2019–2025.

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. (1955). The Frequency Distribution of the Annual Maximum (or Minimum) of Meteorological Events. Quarterly Journal of the Royal Meteorological Society 81, 158–171.

Vogel, R.M. (1986). The Probability Plot Correlation Coefficient Test for the Normal, Lognormal, and Gumbel Distributional Hypotheses. Water Resources Research 22(4), 587–590. (Correction, Water Resources Research 23(10), 2013, 1987.)

See Also

GEVD, egevd, EVD, eevd, Goodness-of-Fit Tests, htest.object.

Examples

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  # Generate 25 observations from a generalized extreme value distribution with 
  # parameters location=2, scale=1, and shape=1, and test the null hypothesis 
  # that the shape parameter is equal to 0. 
  # (Note: the call to set.seed simply allows you to reproduce this example.)

  set.seed(250) 

  dat <- rgevd(25, location = 2, scale = 1, shape = 1) 

  zTestGevdShape(dat) 

  #Results of Hypothesis Test
  #--------------------------
  #
  #Null Hypothesis:                 shape = 0
  #
  #Alternative Hypothesis:          True shape is not equal to 0
  #
  #Test Name:                       Z-test of shape=0 for GEVD
  #
  #Estimated Parameter(s):          shape = 0.6623014
  #
  #Estimation Method:               Unbiased pwme
  #
  #Data:                            dat
  #
  #Sample Size:                     25
  #
  #Test Statistic:                  z = 4.412206
  #
  #P-value:                         1.023225e-05

  #----------

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

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