# Maximum Likeilhood estimate of shape parameter of GPD is negative, even though exceedances are positively skewed

I am looking at fitting a Generalized Pareto Distribution (GPD) to extreme events which exceed a certain value threshold for Bilbao waves data.

Selecting threshold at c=7.5, resulting in 154 exceedances (X-c) and I have used used POT package from R-software, fit GPD for exceedances.

mle=fitgpd(x,u,est="mle")\$param


which resulted in following estimates

          t= 7.5 and m = 156      t= 8 and m = 106
Methods    Scale      Shape        scale      Shape
MLE       1.8602    -0.7681       1.6431    -0.8619
PICK      1.5461    -0.4854       1.2557    -0.4815
Zhang     1.7223     0.6860       1.4618     0.7314


Skewness and histogram reveals that, exceedances are positively skewed, but Maximum likelihood estimate of shape is negative.

I have used other packages like ismev, evir ... al well, but my estimates of shape are still negative.

Can any one help me understand, why am I getting negative value for the shape?

Following is the Bilbao waves data the zero-crossing hourly mean periods (in seconds), above7 seconds, of the sea waves:

7.05 7.26 7.46 7.59 7.69 7.82 7.90 7.97 8.11 8.21 8.40 8.51 8.69 8.85 9.06 9.23 9.46 9.75 9.12 9.24 9.47 9.78 9.16 9.27 9.59 9.79 9.43 9.74 7.12 7.27 7.46 7.59 7.72 7.83 7.91 7.99 8.12 8.23 8.41 8.52 8.71 8.86 7.15 7.28 7.47 7.61 7.72 7.83 7.93 8.00 8.15 8.23 8.42 8.53 8.72 8.88 7.18 7.30 7.48 7.63 7.72 7.83 7.93 8.03 8.15 8.30 8.43 8.54 8.74 8.88 9.17 9.29 9.59 9.79 9.17 9.30 9.60 9.80 9.18 9.32 9.61 9.84 9.22 9.90 7.19 7.31 7.48 7.65 7.72 7.84 7.93 8.03 8.15 8.30 8.43 8.56 8.74 8.94 7.20 7.31 7.52 7.66 7.72 7.85 7.94 8.05 8.18 8.31 8.45 8.58 8.74 8.98 7.20 7.32 7.54 7.66 7.77 7.85 7.95 8.06 8.18 8.31 8.48 8.59 8.74 8.98 7.20 7.33 7.55 7.67 7.77 7.88 7.95 8.06 8.18 8.32 8.49 8.59 8.79 8.99 7.20 7.37 7.55 7.67 7.79 7.88 7.97 8.07 8.19 8.32 8.50 8.60 8.81 9.01 7.25 7.40 7.58 7.68 7.79 7.90 7.97 8.10 8.20 8.33 8.50 8.65 8.84 9.03 9.33 9.62 9.85 9.18 9.36 9.63 9.89 9.21 9.38 9.66

• The POT package offers functions for model checking. How well does the model fit the data? Could you add the diagnostic plots to your question? Commented Jun 15, 2018 at 15:14

Using the POT parameterization, location is $$\mu$$, scale is $$\sigma$$, and shape is $$\xi$$, and it follows the parameterization found here. You find cases where the shape of the GPD is negative when the observations are clustered in a small area. More precisely, when the shape is negative, the support is between 0 and "location - scale/shape" which is location + abs(scale/shape). In your case, your observations are very clustered between 7 and 9. Given what you posted, scale/shape is very roughly between 1.5 and 3 or so, I'm not sure what was picked for location, but the results make sense. Thick tailed distributions which extend across orders of magnitude tend to have a positive shape parameter.