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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 $\phi_\alpha (x)=\alpha \, x_\min^\alpha x^{-\alpha -1}\mathbb{1}_{x\geq x_\min }$;

Looking for 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 $1 < \alpha \leq 2$.

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If $X_1,\ldots,X_n$ are independent, then the even that $X_i>X_{i-1}$ for all $i$ is somewhat improbable. Did you you mean you want to find conditional probabilities given that that happens, or did you mean you want to take independent realizations and then sort them into decreasing order? Or something else? If you wanted to take independent realizations and then sort them into decreasing order, a standard notation for that would be $X_{(n)}>X_{(n-1}>\cdots>X_{(i)}>X_{(i-1)}>\cdots$. ${}\qquad{}$ –  Michael Hardy Apr 20 at 15:24
    
Thanks Michael I fixed the question to show that I order after. Is it fine this way? –  Nero Apr 20 at 16:13

1 Answer 1

$\hat{\kappa}$ 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.

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Thanks, will try. –  Nero Apr 21 at 13:23
    
Is $Z_1$ the max? Taking $\frac{Z_1}{\Sigma Z_i}$ is equivalent to the distribution of max over average? If so, $Z_1$ follows an extreme value distribution and pulling out the ratio doesn't seem problematic. –  Nero Apr 21 at 13:32

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