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I have been able to solve part i of the question below. However I have been unable to do the proof in part ii and iii. I have tried using convolutions/mgfs but i have not been able to obtain the pdf. I would appreciate if anyone could help/guide me to solve this problem. Thanks

Let X and Y be independent positive random variables. We are interested to find a decreasing function T: (0,∞)→(0,∞) such that T(X+Y) is independent of T(X)−T(X+Y). Such a function indeed exists with X being a Generalised Inverse Gaussian (GIG) distribution and Y a gamma distribution and T(x) = 1/x. This property is called Matsumoto-Yor property in the literature.Denote the density of GIG(μ,a,b) random variable by

f(x;μ,a,b)= (1/K(μ,a,b))x^(μ-1)e^(-(a^2x^(-1)+b^2x)/2),μ∈R, a,b >0, x >0

where K(μ,a,b) is a constant depending only on μ,a,b. Denote the density of a gamma random variable γ(μ,a) by

g(y;μ,a) =(a^μ/Γ(μ))y^(μ−1)e^(−ay), μ,a >0,

where Γ(μ) is the Gamma function. Let T(x) = 1/x,x >0.

(i) [3] Let X be a GIG(μ,a,b) random variable. Show that T(X) is distributed as GIG(−μ,b,a).

(ii) [4] If X∼GIG(−λ,a,a) and Y∼γ(λ,a^2/2) are independent random variables, show that X and T(X+Y) have the same distribution.

(iii) [5] Let X and Y be two independent random variables such that X∼GIG(−μ,a,b) and Y∼γ(μ,b^2/2),μ,a,b >0. Show that T(X+Y) is independent of T(X)−T(X+Y). Identify the distributions of T(X) and T(X)−T(X+Y)

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  • $\begingroup$ Can anyone help me? Thanks $\endgroup$
    – Maxtan200
    Aug 12, 2017 at 21:33

1 Answer 1

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Lol, you and I have the same class, STAT$7004$, right? This one's a real doozy. I'm too lazy to type math on a website I don't even have an account for, so let me just give you a nudge in the right direction. Let $U=T(X+Y)$ and $V=T(X)-T(X+Y)$, can you think of a transformation function that gives $(X, Y)$ in terms of $(U, V)$? Once you've done that, it'll only take a simple change of variables to get the joint density of $U$ and $V$. Remember that $U$ and $V$ are independent, and so are $X$ and $Y$, this piece of information is important. Now I think I've given you enough to work out the rest. Say hi to me the next time you see me in class. I usually sit at the back and often have a dark blue Red Bull ball cap on. The name is Bill.

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    $\begingroup$ You guys should get together for a love-fest in the library. $\endgroup$
    – wolfies
    Aug 13, 2017 at 17:09

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