Support for this manuscript was provided by the National Science Foundation (NSF) through the Regional Climate Uncertainty Program (RCUP) at the National Center for Atmospheric Research (NCAR). NCAR is sponsored by NSF and managed by the University Corporation for Atmospheric Research.
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The GP distribution is a conditional distribution with the condition that X > u. Therefore, it does not have a location parameter.
When incorporating covariate information into the parameters of the GEV distribution function, it is important to first allow the location parameter to vary. If the inclusion of the covariate term is found to be significant, then inclusion of covariates in the scale parameter can be tested. Generally, it is desirable not to include covariates in the shape parameter, but if there is reason to do so, then they should be included only after including them with the scale parameter. The issue is for any location-scale family of distributions and not just extreme-value distribution functions. Consider, for example, a normal distribution where the mean is also a location parameter and the standard deviation is also a scale parameter. Because the standard deviation involves deviations about the mean, it follows that incorrect specification of the mean, e.g., ignoring a trend in the mean, will be problematic for estimating the standard deviation. This issue is related to one that arises in polynomial regression where it is well know that fitting a second-order polynomial without any linear term is problematic.
Because of the three types of tail behavior for the extreme-value distributions, with one the heavy-tail case, it is problematic to perform resampling from the data without accounting for the uncertainty in type of tail. The parametric bootstrap avoids this issue.