minimum order statistic estimator, denoted by $\hat{x}^{p_e}_{\mathrm{min}}(y_{1:N})$. Let $\left(y^{(m)}_{1:N}\right)_{m=1}^{N}$ denote the ordered sequence obtained from sorting $y_{1:N}$ in an ascending order, $\hat{x}^{p_e}_{\mathrm{min}}(y_{1:N})$ is defined as $$\begin{aligned}
    \hat{x}^{p_e}_{\mathrm{min}}(y_{1:N}) = y_{1:N}^{(1)} \triangleq \min_{k} y_k.\end{aligned}$$ Noting that for any generic estimator $\hat{x}$ the MSE is given by $$\begin{aligned}
\label{eq:mse_generic}
{\mathrm{MSE}}(\hat{x}) = {\mathrm{Var}}(\hat{x}) + b^2(\hat{x}).\end{aligned}$$ The MSE for BLUE and MVU or any other bias compensated estimator coincides with the estimator’s variance. In case of $\hat{x}^{p_e}_{\mathrm{min}}$, the existing bias enters the MSE.

In order to find the MVU estimator, the first step is to find the PDF $f(y_{1:N};\theta)$ with $\theta$ denoting the parameters of the distribution. If the PDF satisfies regularity conditions, the CRLB can be determined. Any unbiased estimator that satisfies CRLB is thus the MVU estimator. However, the considered PDFs do not satisfy the regularity conditions, $$\begin{aligned}
{\mathbb{E}}\left[\frac{\partial\ln f(y_k;\theta)}{\partial \theta}\right]\neq0.\end{aligned}$$ Hence, the CRLB approach is not applicable. Instead, we rely on the RBLS theorem [@article:IJS_lehmann_1; @article:IJS_lehmann_2; @book:ET_kay_93], to find the MVU estimator. The RBLS theorem [@article:IJS_lehmann_1] states that for any unbiased estimator $\tilde{x}$ and sufficient statistics $T(y_{1:N})$, $\hat{x}={\mathbb{E}}(\tilde{x}\mid T(y_{1:N}))$ is unbiased and ${\mathrm{Var}}(\hat{x})\leq{\mathrm{Var}}(\tilde{x})$. Additionally, if $T(y_{1:N})$ is complete, then $\hat{x}$ is MVU.

As shown in [@book:ET_kay_93], if the dimension of the sufficient 