the density that did not exist…

On Cross Validated, I had a rather extended discussion with a user about a probability density

f(x_1,x_2)=\left(\dfrac{x_1}{x_2}\right)\left(\dfrac{\alpha}{x_2}\right)^{x_1-1}\exp\left\{-\left(\dfrac{\alpha}{x_2}\right)^{x_1} \right\}\mathbb{I}_{\mathbb{R}^*_+}(x_1,x_2)

as I thought it could be decomposed in two manageable conditionals and simulated by Gibbs sampling. The first component led to a Gumbel like density

g(y|x_2)\propto ye^{-y-e^{-y}} \quad\text{with}\quad y=\left(\alpha/x_2 \right)^{x_1}\stackrel{\text{def}}{=}\beta^{x_1}

wirh y being restricted to either (0,1) or (1,∞) depending on β. The density is bounded and can be easily simulated by an accept-reject step. The second component leads to

g(t|x_1)\propto \exp\{-\gamma ~ t \}~t^{-{1}/{x_1}} \quad\text{with}\quad t=\dfrac{1}{{x_2}^{x_1}}

which offers the slight difficulty that it is not integrable when the first component is less than 1! So the above density does not exist (as a probability density).

What I found interesting in this question was that, for once, the Gibbs sampler was the solution rather than the problem, i.e., that it pointed out the lack of integrability of the joint. (What I found less interesting was that the user did not acknowledge a lengthy discussion that we had previously about the Gibbs implementation and that he erased, that he lost interest in the question by not following up on my answer, a seemingly common feature of his‘, and that he did not provide neither source nor motivation for this zombie density.)

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