Find the PDF from quantiles I have been presented a problem of this kind: 
suppose I know the values of k quantiles for a continuous random variable $X$
$$X_{1\%} = x_1, X_{5\%} = x_2, \dots , X_{99\%} = x_{k}$$
so that
$$ F_X(x_1)=1\%, F_X(x_2)=5\%, \dots, F_X(x_k)=99\%  $$
From these informations I want to draw the chart of the PDF.
I thought that I could proceed this way:


*

*interpolate $F_X(x)$ to get a smooth CDF (for instance spline interpolation)

*find the derivative (numerical) of the smoothed CDF at some points to obtain the PDF.


Are there other more direct methods to address this problem? Do you think my solution is solid?
Thank you.
 A: Your approach is valid, if you use the integral of a cubic B-spline to interpolate the quantiles with the condition that all B-spline coefficients are nonnegative.
https://en.m.wikipedia.org/wiki/B-spline
This preserves the monotonicty of the quantiles in the interpolating function, while normal cubic splines do not.
There is no need then to numerically differentiate the spline, you get the PDF from the normal evaluation of the B-spline. The integral of the B-spline curve is also available without numerical computation. You get it from any serious library that implements splines, for example, in Python's scipy module.
The remaining question is how to find the B-spline coefficients in this case. I don't have perfect answer right now, although an efficient algorithm should be possible. A simple brute-force way would be to minimize the squared residuals between the integrated spline and the quantiles with a numerical minimizer, while constraining the coefficients to be nonnegative.
I can add a numerical example in a few days if desired.
