EnvStats (version 2.1.0)

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

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

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

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

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).

Value

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

Details

Let $\underline{x} = x_1, x_2, \ldots, x_n$ be a vector of $n$ observations from a generalized extreme value distribution with parameters location=$\eta$, scale=$\theta$, and shape=$\kappa$. Furthermore, let $\hat{\kappa}_{pwme}$ denote the probability-weighted moments estimator (PWME) of the shape parameter $\kappa$ (see the help file for egevd). Then the statistic $$z = \frac{\hat{\kappa}_{pwme}}{\sqrt{0.5633/n}} \;\;\;\;\;\; (1)$$ is asymptotically distributed as a N(0,1) random variable under the null hypothesis $H_0: \kappa = 0$ (Hosking et al., 1985). The function zTestGevdShape performs the usual one-sample z-test using the statistic computed in Equation (1). The PWME of $\kappa$ 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 $\alpha$-level is adequately maintained for samples as small as $n = 25$.

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

# 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)