Distribution of the elemental regression weights with t-distributed co-variate measurement errors

In this article, a heuristic approach is used to determined the best approximate distribution of $\dfrac{Y_1}{Y_1 + Y_2}$, given that $Y_1,Y_2$ are independent, and each of $Y_1$ and $Y$ is distributed as the $\mathcal{F}$-distribution with common denominator degrees of freedom. The proposed approximate distribution is subject to graphical comparisons and distributional tests. The proposed distribution is used to derive the distribution of the elemental regression weight $\omega_E$, where $E$ is the elemental regression set.