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I have to random variables expressed as $1 \times 1000$ vectors. One of the vectors $B$ is Beta distributed while the other $L$ is lognormal distributed. Upon element-wise multiplication, I get vector $M$ whose distribution I'd like to know.
I've done some good of fit tests, and some tests (Kolmogorov) indicate that $M$ it's a log-logistic, another (AIC) inverse-gaussian. The lognormal doesn't look bad either.
Any ideas, of there's any analytic result out there?
I will assume your two variables are independent (you didn't specify.) A tool for investigating the distribution of the product of independent random variables is the Mellin transform. The Mellin transform of the product is the product of the Mellin transforms of the two factors. I will, as a start, just look at a simplified case. I will look at only a standard lognormal variable $X$ (that is, exponential of standard normal). Its Mellin transform is the moment generating function of $\log X$, so is $M_X(s)=\exp(s^2/2)$. For the Beta random variable $Y$, I will assume that $\beta=1$ in the standard parametrization. Then one can show that $-\log Y$ has an exponential distribution with rate $\alpha>0$. Its moment generating function is then $s \mapsto \alpha /(\alpha-s),\quad s<\alpha$.
Summary: The Mellin transform of $X$ is $M_X(s)=\exp(s^2/2)$, while the Mellin transform of $1/Y$ is $M_{1/Y}(s)=\frac{\alpha}{\alpha-s}$, so $M_Y(s)=\frac{\alpha}{\alpha+s},\quad s<\alpha$.
Then we can conclude that the Mellin transform for the product variable $XY$ is $$ M_{XY}(s)=\frac{\alpha}{\alpha+s} \exp(s^2/2).$$ We could continue by trying to invert this transform analytically, or find an approximate, analytic approximation via for example saddlepoint methods.
For now I'm out of time (beach waiting), will come back.
The PDF of a Beta with parameters (A,B,C,D) is:
$$ \frac{(B − A)^{1-C-D}}{Beta(C,D)}(y-A)^{C-1}(B − y)^{D−1} $$
The PDF of a Lognormal with parameters (A,B) is:
$$ \frac{1}{By\sqrt{2\pi}} exp \bigg[−\frac{1}{2}\left(\frac{log(y) − A}{B}\right)^2 \bigg] $$
(based on this compendium of distributions).
If you spend some minutes looking at all the distributions there, or at those with infinite support here, you can see there is no distribution which PDF has the combination of the above functional forms, including the ones you mention. Although this not really proves that no such distribution exists, I think it is safe at the moment to conclude that there is no analytic characterisation of it in the literature.
