# Weibull distribution fit

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?

Edit: As indicated by the valuable comments, I was indeed looking for a Weibull distribution. I tried fo fit this model using fitdist in R. The results are shown on the figure.

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?

• Would you please post a link to the data? – James Phillips Aug 7 '19 at 16:42
• @user158565 - It's a Weibull distribution. $F_X(0) = 0$. – jbowman Aug 7 '19 at 18:19
• What language are you working with? It makes a difference, as "fitting functions with standard distributions" is a language-specific reference, and your distribution is one of the standard ones (the Weibull.) – jbowman Aug 7 '19 at 18:23
• @user158565 - Ha ha! It's happened to me before too, no worries :). Funny how our minds fail us sometimes... – jbowman Aug 7 '19 at 19:01
• @jbowman I thought I was the only one who did that. – James Phillips Aug 7 '19 at 20:01