is known and then further discuss the unknown hyper parameter case. In the general case, let $\Psi = [x,\beta]^\top$ denote the unknown parameter vector, the Neyman-Fisher factorization gives $h(y_{1:N})=1$ and $$\begin{aligned}
T(y_{1:N}) = \begin{bmatrix}
y^{(1)}_{1:N} \\ y^{(N)}_{1:N}
\end{bmatrix} = \begin{bmatrix}
T_1(y_{1:N}) \\ T_2(y_{1:N})
\end{bmatrix}.
\label{eq:uniform_ss}\end{aligned}$$

### Known hyper parameter $\beta$ {#subsubsec:known_hyper_parameter_uniform}

When the maximum support of the uniform noise $\beta$ is known, the dimensionality of the sufficient statistic is larger than that of the unknown parameter $x$. As discussed in [@book:ET_kay_93], the RBLS theorem can be extended to address this case if the form of a function $g(T_1(y_{1:N}),T_2(y_{1:N}))$ can be found that combines $T_1$ and $T_2$ into a single unbiased estimator of $x$.

Let $Z = T_1(y_{1:N})+T_2(y_{1:N})=u+v$. Since $T_1$ and $T_2$ are dependent,

$$\begin{aligned}
\label{eq:dist_sum_gen}
    f_Z(z) = \int_{-\infty}^{\infty} f_{y^{(1)},y^{(N)}}(u,z-u)\,d_u,
    \end{aligned}$$

where $f_{y^{(1)},y^{(N)}}(u,z-u)$ is the joint density of minimum and maximum order statistics. As shown in [@book:OS_david_04], for $-\infty<u<v<\infty$, the joint density of two order statistics $y^{(i)}$ and $y^{(j)}$ is given by $$\begin{aligned}
    f_{y^{(i)},y^{(j)}}(u,v) = &\frac{N!}{(i-1)!(j-1-i)!(N-j)!}\nonumber\\&\times f_Y(u)f_Y(v)\left[F_Y(u)\right]^{i-1}\nonumber\\
    &\times\left[F_Y(v)-F_Y(u)\right]^{j-1-i}\left[1-F_Y(v)\right]^{N-j},
    \end{aligned}$$ that for the extreme orders, $i=1$ and $j=N$ can be simplified such that for $u<v$ $$\begin{aligned}
\label{eq:order_joint_gen}
    f_{y^{(1)},y^{(N)}}(u,v) = N(N-1)\left[F_Y(v)-F_Y(u)\right]^{N-2} f_Y(u)f_Y(v). 
    \end{aligned}$$ and zero otherwise. Substituting  into , we get $$\begin{aligned}
    f_Z(z) 