Create non-standard PDF from data, then doing a likelihood fit

Question

Let's say I have some set of data which is not a "normal" distribution of data, meaning it does not hold a Gaussian, exponential, or other neat formulaic look. Here's an example of data where the PDF I want to create is based on the solid black line, within the red vertical lines:

Given that I already have the data, how can I create a PDF for this?

Next, I want to compare the data represented by the blue dots, and test whether it will look like the black line if I add some parameter. Say the black line is represented by $F(x)$ and the blue dots are represented by $G(x)$. I want to test:

$G(x) = P1*F(x)$

And I'd like to know what P1 is such that it is the best fit.

Answer 1

If one doesn't know the functional form of a pdf and has a fair amount of data, then a nonparametric density estimate is definitely the way to go. (Finding a parametric density with a parsimonious number of parameters doesn't help if the fit is poor.)

Here is some generated data somewhat similar to what you have above:

data = 18 + 100 RandomVariate[BetaDistribution[1, 1.6], 1000];
Histogram[data, {4}]

A nonparametric density estimate for such bounded data is found as follows:

skd = SmoothKernelDistribution[data, Automatic, {"Bounded", {15, 120}, "Gaussian"}];
Plot[PDF[skd, x], {x, 15, 120}, PlotRange -> {{15, 120}, {0, .015}}]

Another estimated pdf can be constructed from the second set of data and plotted on the same figure. (Much cleaner and useful than attempting to overlay histograms.)