Can anyone confirm the method is correct that I am using to get the truncated Gumbel I extreme value PDF. I am fitting Gumbel I to ice loads that occur on ships, if that is of any help here.
The basic distribution functions.
Gumbel I CDF of the largest value:
$ F_X(x) = exp\left[-e^{-\alpha(x-u)}\right] $
The corresponding PDF:
$ f_X(x) = \alpha_n \cdot e^{-\alpha(x-u)} \cdot exp\left[-e^{-\alpha(x-u)}\right] $
Now why am I uncertain if the truncated extreme value PDF is correct, is due to the differences between how I get the truncated extreme value CDF and the PDF.
First I am going to show how I get the truncated extreme value CDF.
What I do is I calculate the left-truncated CDF. Let's say there is a load value "b" from which lower values I don't consider, then the left-truncated CDF is:
$ F_{trunc}(x) = \frac {F_X(x)-F_X(b)}{1-F_X(b)} $
Then I calculate the trucnated extreme value CDF, which is defined as:
$ G_{trunc}(x) = [F_{trunc}(x)]^N $
, where N is the number of ice load observation during a ship's lifetime (e.g. ship lifetime 25 years, 20 days in ice per year, maximum ice loads during 12 hour period i.e. 2 observations per day. Then N = 25*20*2 = 1000)
Secondly, this is how I get the truncated extreme value PDF.
Here I firstly define the extreme value CDF, which is:
$ G(x) = [F_X(x)]^N $
Then I define the extreme value PDF, which is:
$ g(x) = \frac{\partial G(x)}{\partial x} = \alpha \cdot N \cdot exp\left[-e^{-\alpha(x-u)}\right] \cdot e^{-\alpha(x-u)} \cdot \left[exp[-e^{-\alpha(x-u)}]\right]^{N-1} $
And finally I truncate the extreme value PDF as follows:
$ g_{trunc} = \frac {g(x)}{1-G(b)} $
You see that for the CDF I firstly calculate the truncated distribution and then with the truncated distribution I calculate the extreme value distribution for CDF (which could be also called the long-term distribution for CDF). However, for the PDF I first define the extreme value distribution and then truncate the extreme value distribution to get the truncated extreme value PDF.
