general order statistic $k$ can be obtained by generalizing the results of the previous section, and considering the event $\{y_i \leq y\}, i = 1, 2, \ldots, k$ as a “success,” and letting $\zeta$ = the number of such successes in $N$ mutually independent trials, $$\begin{aligned}
    F_{(k,N)}(y)=\sum_{n=k}^{N-1}\begin{pmatrix}N\\n\end{pmatrix}[F(y)]^n[1-F(y)]^{N-n}+[F(y)]^N.
    \label{eq:cdf_generic2}
    \end{aligned}$$ Differentiating  with respect to $y$ gives a telescoping sum of the form, $$\begin{aligned}
f_{(k,N)}(y)&=\sum_{n=k}^{N-1}\frac{N!}{(n-1)!(N-n)!}[F(y)]^{n-1}f(y)[1-F(y)]^{N-n}\nonumber\\ &+ \sum_{n=k}^{N-1}\frac{N!}{n!(N-n-1)!}[F(y)]^n[1-F(y)]^{N-n-1}(-f(y)) \nonumber\\&+ N[F(y)]^{N-1}f(y),\end{aligned}$$ in which, except the first term, all other terms cancel each other. Hence, the marginal probability density function of the $k$:th order statistic of a set of $N$ independent and identically random variables with common CDF $F(y)$ and PDF $f(y)$ is given by, $$\begin{aligned}
f_{(k,N)}(y) = Nf(y)\begin{pmatrix}N-1\\k-1\end{pmatrix}F(y)^{k-1}\left(1-F(y)\right)^{N-k}.
\label{eq:density_order}\end{aligned}$$

Location Estimation Problem {#sec:location_estimation_problem}
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Consider the location estimation problem in which we have measurements $y_k$, $k=1,\ldots,N$ of the unknown parameter $x$. Assuming that the measurements are corrupted with additive noise $e_k\sim p_e(\theta)$, where $\theta$ denotes the parameter(s) of the noise distribution, the measurement model is given by $$\begin{aligned}
y_k=x+e_k, \quad k=1,\ldots,N.
\label{eq:generic_model}\end{aligned}$$ The BLUE for the estimation problem  is given by $$\begin{aligned}
\hat{x}_{\mathrm{BLUE}}^{p_e(\theta)}(y_{1:N},\theta) &= \frac{1}{N}\sum_{k=1}^N y_k- \delta(\theta),
\label{eq:mean_estimator}\end{aligned}$$ where $y_{1:N} = \{y_k\}_{k=1}^{N}$ and $\delta(\theta)=\mathbb{E}(e_k)$ is the bias compensation term.

In the following sections, closed-form 