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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?

Mark
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2 Answers2

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Before you get to the likelihood function, the first issue here is to confirm that this is a valid cumulative distribution function. If you examine the requirements for this, you will see that you must have $a=1$ for validity of the form, so your distribution will reduce to the two-parameter Weibull distribution.

Once you have narrowed down the CDF, you then obtain the PDF using differentiation and use the latter to form the likelihood function for an arbitrary sample.

Ben
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    So really I just have a non-linear function that to my untrained eye resembled a CDF. Thank you. – Mark Sep 10 '21 at 12:04
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You can get easily get the PDF.

  1. 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.

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

dario
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