I'm trying to curve fit a functional form to observed data - where my observed data is a physical observation that represents a CDF.

Visually - my data points currently look like this:

y_spl_2d = y_spl.derivative(n=1)
plt.plot(x_range,y_spl_2d(x_range), 'ro')

My goal is to get an accurate representation of the PDF. Just a visual inspection of the CDF looks good (I know... not that that means much).

If I take the second derivative of this function numerically there is TONS of noise introduced. This is what it looks like:

This is where I'm at. Can I fit a functional form to this CDF data so I can take a derivative symbolically?

What is the best way - in a statistical sense - to not lose information like bi-modal PDFs and other things like this. How can I include enough paramters in my functional form to include skewness, kurtosis, etc (to not force a perfectly guassian distribution)?

The original fit where all of this data derives from is the derivative of an options price to the strike.  

I'm trying to curve fit a functional form to observed data - where my observed data is a physical observation that represents a CDF.

Visually - my data points currently look like this:

y_spl_2d = y_spl.derivative(n=1)
plt.plot(x_range,y_spl_2d(x_range), 'ro')

My goal is to get an accurate representation of the PDF. Just a visual inspection of the CDF looks good (I know... not that that means much).

If I take the second derivative of this function numerically there is TONS of noise introduced. This is what it looks like:

This is where I'm at. Can I fit a functional form to this CDF data so I can take a derivative symbolically?

What is the best way - in a statistical sense - to not lose information like bi-modal PDFs and other things like this. How can I include enough parameters in my functional form to include skewness, kurtosis, etc (to not force a perfectly Gaussian distribution)?

The original fit where all of this data derives from is the derivative of an options price to the strike. 

I'm trying to curve fit a functional form to observed data - where my observed data is a physical observation that represents a CDF.

Visually - my data points currently look like this:

y_spl_2d = y_spl.derivative(n=1)
plt.plot(x_range,y_spl_2d(x_range), 'ro')

My goal is to get an accurate representation of the PDF. Just a visual inspection of the CDF looks good (I know... not that that means much).

If I take the second derivative of this function numerically there is TONS of noise introduced. This is what it looks like:

This is where I'm at. Can I fit a functional form to this CDF data so I can take a derivative symbolically?

What is the best way - in a statistical sense - to not lose information like bi-modal PDFs and other things like this. How can I include enough paramters in my functional form to include skewness, kurtosis, etc (to not force a perfectly guassian distribution)?

I'm trying to curve fit a functional form to observed data - where my observed data is a physical observation that represents a CDF.

Visually - my data points currently look like this:

y_spl_2d = y_spl.derivative(n=1)
plt.plot(x_range,y_spl_2d(x_range), 'ro')

My goal is to get an accurate representation of the PDF. Just a visual inspection of the CDF looks good (I know... not that that means much).

If I take the second derivative of this function numerically there is TONS of noise introduced. This is what it looks like:

This is where I'm at. Can I fit a functional form to this CDF data so I can take a derivative symbolically?

What is the best way - in a statistical sense - to not lose information like bi-modal PDFs and other things like this. How can I include enough paramters in my functional form to include skewness, kurtosis, etc (to not force a perfectly guassian distribution)?

The original fit where all of this data derives from is the derivative of an options price to the strike.