
CMEName:
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}} The CME, or mean residual life (MRL), is the expectation of the amount by which a value exceeds a threshold u, conditional on that threshold being attained. If the exceedance data are fitted by the GPD model and c < 1, u > 0, and (a + u*c) > 0, then a plot of CME versus u should follow a line with intercept a/(1c) and slope c/(1c). The linearity of the CME plot can thus be used as an indicator of the appropriateness of the GPD model and both c and a can be estimated. 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 CME command performs a least squares fit of the CME versus u data points. It does this as follows:
The NISTIR 5531 document (see the References section below) gives the formula for the standard deviation of c.
where <y> is the response variable; and where the <SUBSET/EXCEPT/FOR qualification> is optional.
CME 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.
Heckert, Simiu, and Whalen (1998), "Estimates of Hurricane Wind Speeds by the "Peaks Over Threshold" Approach," Journal of Structural Engineering. Simiu and Heckert (1996), "Extreme Wind Distribution Tails: A "Peaks Over Threshold" Approach," Journal of Structural Engineering. Lechner, Simiu, and Heckert (1993), "Assessment of 'peak over threshold' Methods for Estimating Extreme Value Distribution Tails," Structural Safety. Simiu, Heckert, and Whalen (1996), "Estimates of Hurricane Wind Speeds by the 'Peaks Over Threshold' Method," NIST Technical Note 1416. Gross, Simiu, Heckert, and Lechner (1995), "Extreme Wind Estimates by the Conditional Mean Exceedance Procedure," NISTIR 5531. Simiu and Heckert (1995), "Extreme Wind Distribution Tails: A 'Peaks Over Threshold' Approach," NIST Building Science Series 174.
2005/5: Added support for HTML and Latex output. 2005/5: Added support for the standard deviation of c. SKIP 25 READ MPOST550.DAT Y SET WRITE DECIMALS 5 LET Y2 = SORT Y LET THRESHOL = Y2(900) SET WRITE DECIMALS 5 CME MLE YThe following output is generated. Generalized Pareto Parameter Estimation (CME) (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 CME Parameter Estimates: Location Parameter: 43.36000 Scale Parameter: 16.24223 Shape Parameter (Gamma): 0.20209 Standard Deviation of Gamma: 0.05361 Loglikelihood: 0.8068094E+02 AIC: 0.1673619E+03 AICc: 0.1676907E+03 BIC: 0.1743933E+03 For negative Gamma, the generalized Pareto is equivalent to a reverse Weibull (SET MINMAX MAX) with: Shape Parameter (Gamma): 4.94840 Location Parameter: 101.72727 Scale Parameter: 48.99755  
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Date created: 06/05/2001 