How do you test a distribution?
Well you have a set of data. We start by sorting the data. The lowest
value has no values below it, so it gets the value (Xl,0). The highest
value has no values above it, so it gets the value of (Xh,1). All the
other data points are now (Xi,fraction of way between Xl and Xh).
You now plot (Xi,Yi). In general, you get some kind of sigmoid (S
shaped) curve. It is monotone increasing.
You could smooth that curve (if you think the distribution is smooth).
If you have reason to believe your data (X,Y) is exact, you could fit a
cubic spline to the data and specify that the slope at 0 is 0, and the
slope at 1 is 0. That will probably introduce a little wiggle to the
spline fit, since we really only have slopes of 0 at the extremities if
the X variable, and not necessarily at the extremities of our sampled
data. The cubic spline I was first taught, is fitted by solving a
linear system for all the data points at one time. This means a little
error in one data point affects all parameters calculated. Which often
leads to wiggle. Some splines are "localised", the Akima spline is one
such (family of) spline.
If you know something about the error in your data, you could calculate
a smoothing spline through the data.
In any event there are lots of choices as to how to analyze things.