I am looking at the possibility of using the Wakeby Distribution to attempt to model color components in image rows and columns (it is a very silly idea for "compressing" images that I want to see explode). I chose this distribution because I like how it seems possible to model a PDF that seems to respect skewness (naturally, any comments about distributions that can be parameterized using moments like mean, variance, skewness, and kurtosis would be appreciated). I think I can ramp up on the MLE method to estimate $\alpha$, $\beta$, $\gamma$, and $\delta$ if I can just get past this one problem.
Both the Wikipedia page and the NIST page on the Wakeby Distribution state that finding the PDF for the distribution requires numerically inverting the "percent point function" without any further explanation. From other research, I have found that "percent point function" is another name for the cumulative density function that is referred to as $U$ on both those pages. In trying to invert the formula to solve for $U$, I have gotten as far as
$\beta\delta(x-\xi)+\beta\gamma-\alpha\delta=\frac{\beta\gamma-\alpha\delta(1-U)^{\beta+\delta}}{(1-U)^\delta}$
but I can get no further. I have tried looking for any sites or papers that address this issue, but it seems that everyone uses external tools to perform this work.
Could someone help me understand how to find the cumulative distribution function $F(x)$ from the Wakeby Distribution formula?
I am a layman, but it seems that you have to solve the integral (your function in the integral boundaries (in the given interval)). A possible way to do it, is numerical integration (e.g. simpson's rule).