
DEHAANName:
The cumulative distribution function of the generalized Pareto distribution is
Here, c is the shape parameter and a is the scale parameter. This equation can be used to represent the conditional cumulative distribution of the excess Y = X  u of the variate X over the threshold u, given X > u for u sufficiently large. The cases c > 0, c = 0, and c < 0 correspond respectively to the extreme value type II (Frechet), extreme value type I (Gumbel), and reverse Weibull domains of attraction. Given the mean E(Y) and standard deviation s_{Y} of the variate Y, then
c = 0.5*{1  [E(Y)/s_{Y}]^{2}} Note that for the case where c < 0, then \( \gamma\) = 1/c is the estimate of the shape parameter for the reverse Weibull (SET MINMAX 2 case in Dataplot) distribution. The de Haan estimates of a and c are determined as follows.
where <y> is the response variable; and where the <SUBSET/EXCEPT/FOR qualification> is optional.
DEHAAN MLE Y SUBSET TAG > 0
If no threshold is specified, then the minimum data value is used as the threshold.
If the absolute value of GAMMA is within a userspecified tolerance of zero, then the following are also saved.
To specify this tolerance, enter the command
The default tolerance is 0.05. If GAMMA is less than zero with an absolute value greater than the above tolerance, then the following are also saved.
These estimates for the reverse Weibull and Gumbel distributions are based on moment estimators. The formulas are given on page 3 of NIST Building Science Series 174 (see the Reference section below). Currently, no estimates for the Frechet case (GAMMA > 0) are saved.
for details. The ASCII output was also modified somewhat. This was a cosemetic change to make the output clearer.
"Estimates of Hurricane Wind Speeds by the "Peaks Over Threshold" Approach", Alan Heckert, Emil Simiu, and Tim Whalen, Journal of Structural Engineering, April, 1998. "Extreme Wind Distribution Tails: A "Peaks Over Threshold" Approach", Simiu and Heckert, Journal of Structural Engineering, May, 1996. "Assessment of 'peak over threshold' Methods for Estimating Extreme Value Distribution Tails", J. A. Lechner, E. Simiu, N. A. Heckert, Structural Safety, 1993. "Estimates of Hurricane Wind Speeds by the 'Peaks Over Threshold' Method", E. Simiu, N. A. Heckert, T. Whalen, NIST Technical Note 1416, February, 1996. "Extreme Wind Estimates by the Conditional Mean Exceedance Procedure", J. L. Gross, E. Simiu, N. A. Heckert, J. A. Lechner, NISTIR 5531, April, 1995. "Extreme Wind Distribution Tails: A 'Peaks Over Threshold' Approach", E. Simiu, N. A. Heckert, NIST Building Science Series 174, March, 1995
2005/5: Added support for HTML/Latex format output. SKIP 25 READ MPOST550.DAT Y LET Y2 = SORT Y LET THRESHOL = Y2(900) SET WRITE DECIMALS 5 DEHAAN MLE YThe following output is generated. Generalized Pareto Parameter Estimation (de Haan) (Maximum Case) Summary Statistics (Full Data Set): Number of Observations: 977 Sample Mean: 7.81898 Sample Standard Deviation: 17.76409 Sample Minimum: 0.00000 Sample Maximum: 90.04000 Summary Statistics for Observations Above Threshold: Threshold: 43.36000 Number of Observations Above Threshold: 77 Sample Mean: 56.76623 Sample Standard Deviation: 10.39647 de Haan Parameter Estimates: Location (Threshold) Parameter: 43.36000 Scale Parameter: 14.92411 Shape Parameter (Gamma): 0.35375 Standard Deviation of Gamma: 0.13108 For negative Gamma, the generalized Pareto is equivalent to a reverse Weibull (SET MINMAX MAX) with: Shape Parameter (Gamma): 2.82687 Location Parameter: 54.16054 Scale Parameter: 52.02372  
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Date created: 6/5/2001 