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

Question

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

Answer 1

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.