Comparing kernel density estimates with polynomial functions

Question

I have kernel density estimates of survival functions from different datasets. These density estimates are produced just to have smoothed versions of the original survival function which is a step function.

I don't know the functional form of the kernel that was used. I need to compare these kernel density estimated curves - is fitting polynomials to these smooth kernel density estimates a good way to see if they differ from each other much?

Its important that I compare the smoothed curves and not the underlying unsmoothed data. I am in a way dealing with the same type of problem as here: Fitting a CDF to differentiate symbolically only I already have the kernel densities, and just need some metrics to compare them. I already posted this question here Polynomial fit for a kernel smoothed function,

but think this is a better way to phrase it.

Answer 1

If you want to compare across different settings, then using polynomials might not help much compared to just comparing the raw kernels at different number of periods out. For instance, there is no simple interpretation of the third order term being larger of smaller. You need to plot them together, I suppose.

Moreover, polynomials can look weird in the end points.