I have a random variable $X$ for which it is known a priori that $P(X > x) = \exp(-ax^b)$, i.e. the CDF is given by $F_X(x) = 1-\exp(-ax^b)$.
I would like to determine the values of $a$ and $b$ but I am currently struggling to find an appropriate transformation of this "exponential-power" distribution that would allow me using the fitting functions with standard distributions.
Would you have any idea of transformation (or any alternative) that would help this purpose?
At first glance, it looks pretty satisfying (except for the tail on the Q-Q plot). However, the Kolmogorov-Smirnov goodness of fit test dramatically fails (p-value < 1e-14). Should I reject the model based on the KS test results?