= \frac{1}{2}N\beta^{-N}(2x+2\beta-z)^{N-1},
    \end{aligned}$$ for $2x+\beta<z<2(x+\beta)$ and $$\begin{aligned}
    f_Z(z) = -\frac{1}{2}N\beta^{-N}\frac{(z-2x)^{N}}{2x -z},
    \end{aligned}$$ for $2x<z\leq2x+\beta$ and zero otherwise. It can be shown that $$\begin{aligned}
    {\mathbb{E}}(f_Z(z)) = 2x + \beta.
    \end{aligned}$$

Hence, noting that $\beta$ is known, the function $g(T_1(y_{1:N}),T_2(y_{1:N}))$ that gives an unbiased estimator should be of the form of

$$\begin{aligned}
    \hat{x}_{\mathrm{MVU}}^{\mathcal{U}}(y_{1:N},\beta) &= g(T_1(y_{1:N}),T_2(y_{1:N}))\nonumber\\
    &= \frac{1}{2}(y^{(1)}_{1:N}+ y^{(N)}_{1:N}) - \frac{\beta}{2}.
    \end{aligned}$$

The MSE of the MVU estimator is given by $$\begin{aligned}
    {\mathrm{MSE}}\left(\hat{x}_{\mathrm{MVU}}^{\mathcal{U}}(y_{1:N},\beta) \right) = \frac{\beta^2}{2N(N+3)+4}.\label{eq:mse_mvu_unif_known}
    \end{aligned}$$

Comparing to , the order statistics based MVU estimator outperforms the BLUE one order of magnitude.

### Unknown hyper parameter $\beta$ {#subsec:unknown_hyper_parameter_uniform}

In this case, the MVU estimators for the parameter vector $\Psi=[x,\beta]^\top$ can be derived from sufficient statistics , $$\begin{aligned}
\hat{\Psi}=g(T(y_{1:N})),\quad \mathrm{s.t.}\quad {\mathbb{E}}\left(g\left(T(y_{1:N})\right)\right) = \Psi.\end{aligned}$$ In this case, we have $$\begin{aligned}
{\mathbb{E}}(T(y_{1:N})) = \begin{bmatrix}
x+\frac{\beta}{N+1}\\\\x+\frac{N\beta}{N+1}
\end{bmatrix}
\label{eq:uniform_unknown_E_gen}\end{aligned}$$ To find the transformation that makes  unbiased, we define

$$\begin{aligned}
    g(T(y_{1:N}))=\begin{bmatrix}\frac{1}{N-1}\left(NT_1(y_{1:N})-T_2(y_{1:N})\right)\\\\   \frac{N+1}{N-1}\left(T_2(y_{1:N})-T_1(y_{1:N})\right)\end{bmatrix}
    \end{aligned}$$

that gives $$\begin{aligned}
    {\mathbb{E}}\left(g(T(y_{1:N}))\right) = \begin{bmatrix}x\\\beta\end{bmatrix}.
    \end{aligned}$$

Finally, the MVU estimator of $x$ when the hyper parameter $\beta$ is unknown is given by

$$\begin{aligned}
    \hat{x}_{\mathrm{MVU}}^{\mathcal{U}}(y_{1:N}) = \frac{N}{N-1}y^{(1)}_{1:N} - \frac{1}{N-1} y^{(N)}_{1:N}.
    \end{aligned}$$

and its MSE is $$\begin{aligned}
    {\mathrm{MSE}}\left(\hat{x}_{\mathrm{MVU}}^{\mathcal{U}}(y_{1:N})\right) = \frac{N\beta^2}{(N+2)(N^2-1)}.
    \end{aligned}$$

This is naturally slightly larger than  for finite $N$.

Distributions in the exponential family {#sec:distributioons_in_the_exponential_family}
