I have a sample of about 1000 values. These data are obtained from the product of two independent random variables ξ∗ψ. The first random variable has a uniform distribution ξ∼U(0,1). The distribution of the second random variable is not known. How can I estimate the distribution of the second (ψ) random variable?
We have, Assuming ψ has support on the positive real line,
ξψ=X Where X∼Fn and Fn is the empirical distribution of the data.
Taking the log of this equation we get,
Thus by Levy’s continuity theorem, and independance of ξ andψ
taking the charactersitic functions:
Given that Ψln(X)=1n∑1000k=1exp(itXk),
With X1...X1000 The random sample of ln(X).
We can now specify completly the distribution of Log(ψ) through its characteristic function:
If we assume that the moment generating functions of ln(ψ) exist and that t<1 we can write the above equation in term of moment generating functions:
It is enough then to invert the Moment generating function to get the distribution of ln(ϕ) and thus that of ϕ