Difference of Gamma random variables

Given two independent random variables XGamma(αX,βX) and YGamma(αY,βY), what is the distribution of the difference, i.e. D=XY?

If the result is not well-known, how would I go about deriving the result?

Answer

I will outline how the problem can be approached and state
what I think the end result will be for the special case
when the shape parameters are integers, but not fill in the
details.

  • First, note that XY takes on values in (,)
    and so fXY(z) has support (,).

  • Second, from the standard results that the
    density of the sum of two independent continuous random variables is the
    convolution of their densities, that is,
    fX+Y(z)=fX(x)fY(zx)dx
    and that the density of the random variable Y is
    fY(α)=fY(α), deduce that
    fXY(z)=fX+(Y)(z)=fX(x)fY(zx)dx=fX(x)fY(xz)dx.

  • Third, for non-negative random variables X and Y, note that the
    above expression simplifies to
    fXY(z)={0fX(x)fY(xz)dx,z<0,0fX(y+z)fY(y)dy,z>0.

  • Finally, using parametrization Γ(s,λ) to mean a
    random variable with density
    λ(λx)s1Γ(s)exp(λx)1x>0(x),
    and with
    XΓ(s,λ) and YΓ(t,μ) random variables,
    we have for z>0 that
    fXY(z)=0λ(λ(y+z))s1Γ(s)exp(λ(y+z))μ(μy)t1Γ(t)exp(μy)dy=exp(λz)0p(y,z)exp((λ+μ)y)dy.
    Similarly, for z<0,
    fXY(z)=0λ(λx)s1Γ(s)exp(λx)μ(μ(xz))t1Γ(t)exp(μ(xz))dx=exp(μz)0q(x,z)exp((λ+μ)x)dx.


These integrals are not easy to evaluate but for the special case
s=t, Gradshteyn and Ryzhik, Tables of Integrals, Series, and Products,
Section 3.383, lists the value of
0xs1(x+β)s1exp(νx)dx
in terms of polynomial, exponential and Bessel functions of β
and this can be used to write down explicit expressions for fXY(z).


From here on, we assume that s and t are integers so
that p(y,z) is a polynomial in y and z of degree (s+t2,s1)
and q(x,z) is a polynomial in x and z of degree (s+t2,t1).

  • For z>0, the integral (1)
    is the sum of s Gamma integrals with respect to y with coefficients
    1,z,z2,zs1. It follows that the density of
    XY is proportional to a mixture density of
    Γ(1,λ),Γ(2,λ),,Γ(s,λ)
    random variables for z>0. Note that this result
    will hold even if t is not an integer.

  • Similarly, for z<0,
    the density of
    XY is proportional to a mixture density of
    Γ(1,μ),Γ(2,μ),,Γ(t,μ)
    random variables flipped over, that is,
    it will have terms such as (μ|z|)k1exp(μz)
    instead of the usual (μz)k1exp(μz).
    Also, this result will hold even if s is not an integer.

Attribution
Source : Link , Question Author : FBC , Answer Author : Dilip Sarwate

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