$k\in\mathbb{N}^+$, and $k\in[1,N]$ we can the change the factorials to gamma functions, $$\begin{aligned}
    f_{(k,N)}^{\mathcal{U}[0,\beta]}(\tilde{y}) = \frac{\Gamma(N+1)}{\Gamma(k)\Gamma(N-k+1)} \left(\frac{\tilde{y}-x}{\beta}\right)^{k-1}\left(\frac{\beta-(\tilde{y}-x)}{\beta}\right)^{N-k}.
    \label{eq:order_uniform_general}
    \end{aligned}$$

The marginal distribution  is a generalized beta distribution, also known as four parameters beta distribution [@article:Mcdonal_JE_95]. The support of this distribution is from $0$ to $\beta>0$ and $f_{(k,N)}^{\mathcal{U}[0,\beta]}(\cdot)=\frac{1}{\beta}f_{(k,N)}^{\mathcal{U}[0,1]}(\cdot)$. The bias and variance of the general $k$:th order statistic estimator $\hat{x}_{(k)}^{\mathcal{U}}(y_{1:N})$ in case of uniform noise with support on $[0,\beta]$ are given by

\[eq:moments\_uniform\_order\] $$\begin{aligned}
    {\mathbb{E}}(\hat{x}_{(k)}^{\mathcal{U}}(y_{1:N})) &= \frac{\beta k}{N+1},\\
    {\mathrm{Var}}(\hat{x}_{(k)}^{\mathcal{U}}(y_{1:N})) &= \frac{k(N-k+1)\beta^2}{(N+1)^2(N+2)}.
    \end{aligned}$$

The first two moments of the minimum order statistic estimator are obtained by letting $k=1$ in 

\[eq:moments\_uniform\_min\] $$\begin{aligned}
    b\left(\hat{x}_{\mathrm{min}}^{\mathcal{U}}(y_{1:N})\right) &= \frac{\beta}{N+1}\\
    {\mathrm{Var}}\left(\hat{x}_{\mathrm{min}}^{\mathcal{U}}(y_{1:N})\right) &= \frac{N\beta^2}{(N+1)^2(N+2)}.
    \end{aligned}$$

The MSE of $\hat{x}_{\mathrm{min}}^{\mathcal{U}}(y_{1:N})$ is then given by $$\begin{aligned}
{\mathrm{MSE}}\left(\hat{x}_{\mathrm{min}}^{\mathcal{U}}(y_{1:N})\right) = \frac{2\beta^2}{(N+1)(N+2)}.
\label{eq:mse_min_uniform}\end{aligned}$$

MVU estimator {#subsec:mvu_estimator}
-------------

In order to find the MVU estimator, we note that the PDF can be written in a compact form using the step function $\sigma(\cdot)$ as

$$\begin{aligned}
    f(y_k;x,\beta) = \frac{1}{\beta}\left[\sigma(y_k-x) - \sigma(y_k-x-\beta)\right].
    \label{eq:uniform_pdf_single}
    \end{aligned}$$

which gives $$\begin{aligned}
    f(y_{1:N};&x,\beta) = \frac{1}{\beta^N}\prod_{k=1}^{N}\left[\sigma(y_k-x) - \sigma(y_k-x-\beta)\right]\nonumber\\
    &=\frac{1}{\beta^N}\left[\sigma( y^{(1)}_{1:N} -x) - \sigma(y^{(N)}_{1:N}-x -\beta)\right],        \label{eq:unknown}
    \end{aligned}$$

where $y^{(N)}_{1:N}\triangleq\max_k y_k,\quad k=1,\ldots,N$. The expressions for the MVU estimator is derived for two different scenarios. We first assume that the hyper parameter $\beta$ of the noise distribution 