Let us define the Hill estimator by
\begin{align} \widehat{\gamma}_H:=\frac{\int_{X_{n-k,n}}^\infty \log u-\log X_{n-k,n} \: dF_n(u)}{1-F_n(X_{n-k,n})}=\frac n k \int_{X_{n-k,n}}^\infty \log u-\log X_{n-k,n} \: dF_n(u), \end{align} since we denote by $F_n$ the empirical distribution function. $X_{n-k,n}$ is the intermediate order statistic. We can also write: \begin{align} \widehat{\gamma}_H := \frac 1 k \sum_\limits{i=0}^{k-1} \log X_{n-i,n}-\log X_{n-k,n}. \end{align} How can we deduce now, that
$$ \widehat{\gamma}_H = \int_0^1 \log X_{n-\lfloor ks \rfloor, n} - \log X_{n-k,n} \: \mathrm{d}s $$ with $\operatorname{floor}(x)=\lfloor x \rfloor$ holds?
I hope someone can give me a hint or explain it to me.
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