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?

**Answer**

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,

Log(ξ)+Log(ψ)=Log(X)

Thus by Levy’s continuity theorem, and independance of ξ andψ

taking the charactersitic functions:

ΨLog(ξ)(t)ΨLog(ψ)(t)=ΨLog(X)

Now, ξ∼Unif[0,1],therefore−Log(ξ)∼Exp(1)

Thus,

ΨLog(ξ)(−t)=(1+it)−1

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:

(1+it)−1ΨLog(ψ)(t)=1n1000∑k=1exp(itXk)

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:

MLog(ψ)(t)=1n1000∑k=1exp(−tXk)(1−t)

It is enough then to invert the Moment generating function to get the distribution of ln(ϕ) and thus that of ϕ

**Attribution***Source : Link , Question Author : Andy , Answer Author : Drmanifold*