Where $X_{(1)}, X_{(2)}, \ldots,X_{(n)}$ are sorted independents r.v.s, where we index and order in such a way that $X_{(i)} \geq X_{(i-1)}$, $i>1$ where all realizations follow the same Standard Pareto distribution with density $\phi_\alpha (x)=\alpha \, x_\min^\alpha x^{-\alpha -1}\mathbb{1}_{x\geq x_\min }$;
What is the in-sample distribution of the ratio of the ordered sum above the $q^{th}$ largest observation to the total,with a total sample of $n$?
$$ \hat{\kappa}=\frac{X_{(q)}+X_{(q+1)}+\cdots+X_{(n)}}{X_{(1)}+X_{(2)}+\cdots+X_{(n)}}$$
All I have is $0 \leq \hat{\kappa} \leq 1$. Where $\kappa$ is the true value of the estimator, which we were able to derive in closed form, we see biases in Monte Carlo where $\hat{\kappa} < \kappa$ even for large $n$ (at an exponent $\alpha=1.1$ and would like to get an idea of the in-sample distribution. We assume $1 < \alpha \leq 2$.
$\hat{\kappa}$ can be reformulated as an L-statistic.
$\hat{\kappa} = 1 - \frac{X_1+\cdots+X_{q-1}}{X_1+\cdots+X_n}$. Secondly, observe that $\hat{\kappa} \overset{D}{=} 1 - (Y_1 + \cdots + Y_{q-1})$, where $Y_{i}$ is the $ith$ order statistic for an iid sample where each sample follows the same law as $\frac{Z_1}{Z_1+\cdots +Z_n}$, where $Z_i$ are iid Pareto.
There are some references to this problem in the Order Statistics book by David and Nagaraja in chapters 6 and 11. Exact expressions might be tough, but there are asymptotics in chapter 11.