How do I determine the appropriate likelihood function for a custom CDF I'm doing some work describing how things dissolve in solution, and I've determined that a particular parameterization of the 3P Weibull provides a good fit. Now what I want to do is specify target amounts dissolved at a particular time and use MLE to find the values for the parameters. Since there isn't just one 3-parameter Weibull distribution, I can't just turn to R or Python to get their pre-defined distributions. Instead, I need to figure out how to determine the likelihood function (negative log likelihood?) that I should pass into a minimizing function.
The CDF I'm using is $a * (1-e^{-(x/b) ^ c})$ where a, b, and c are the parameters and x represents an amount of elapsed time.
Is there any direction on how I go from this CDF (I don't know the PDF) to a likelihood function? Do I even need to assume this is a distributional problem? Is there a better way to do what I'm trying to do?

You can get easily get the PDF.

*

*If your values are in the range $0$ to $+\infty$ then you should have $a=1$; this because CDF goes to $1$ as $x$ goes to infinity.


*The PDF is simply the derivative of the CDF,  it is $f(x)=\frac{a*c}b*\left({\frac{x}b}\right)^\left(c-1\right)*e^{-\left(\frac{x}b\right)^c}$
