To clarify, I have a discrete value of an unknown distribution. The value of this distribution domain is from 0 to 1. But the value which the discrete value is sampled has an unknown interval.
For example if I have a function of
f(x) = x. And the data I have the f(x) value of
[0.1, 0.7, 1.0], so the x value were sampled at
[0.1, 0.7, 1.0] respectively. But assume I don't know the x value. So how can I approximate the f function, or at least approximate the
x value for each
f(x) on an unknown function?
The incorrect way that I can think of is to use
x value evenly anyway. So the paired value will be
(0, 0.1), (0.5, 0.7), (1.0, 1.0) and then use assumption what kind of
f(x) function would be. In this case is a linear function. So I can solve for least squared for that.
Some floating idea is to use neuron network to approximate
f(x). And somehow use
KL-divergence with the output (but I don't know how).
But is there a theory in statistics that deal with these kind of problem? If so, please let me know
Thank you in advance