Product of two independent random variables

I have a sample of about 1000 values​​. These data are obtained from the product of two independent random variables ξψ. The first random variable has a uniform distribution ξU(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 XFn 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, ξUnif[0,1],thereforeLog(ξ)Exp(1)
Thus,
ΨLog(ξ)(t)=(1+it)1

Given that Ψln(X)=1n1000k=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)=1n1000k=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)=1n1000k=1exp(tXk)(1t)

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

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