in this case, it is not possible to find an unbiased estimator when the hyper parameters $\alpha$ and $\beta$ are unknown. In case of known hyper parameters, the unbiased minimum order statistic estimator, however, can be computed. The unbiased estimator based on minimum order statistic is given by, $$\begin{aligned}
    \hat{x}^{\mathrm{Weibull}}(y_{1:N},\beta,\alpha) &= y^{(1)}_{1:N}- \beta N^{-\frac{1}{\alpha}}\Gamma(1+\frac{1}{\alpha}),
    &\nonumber\\ 
    {\mathrm{MSE}}(\hat{x}^{\mathrm{Weibull}}(y_{1:N},\beta,\alpha)) &=\beta^2N^{\frac{-2}{\alpha}}\left[\Gamma(\frac{\alpha+2}{\alpha})-\left(\Gamma(\frac{\alpha+1}{\alpha})\right)^2\right].
    \end{aligned}$$ An order-statistics-based unbiased estimator with unknown hyper parameters of the distribution could not be obtained.

Other Distributions {#sec:other_distributions}
===================

In this section, we further study the location estimation problem for two other noise distributions. In the rest, the Pareto distribution with positive support is first studied followed by the mixture of uniform and normal distribution.

Pareto distribution
-------------------

Let the scale parameter $\beta$ (necessarily positive) denote the minimum possible value of $y_k$, and $\alpha>0$ denote the shape parameter. The Pareto Type I distribution is characterized by $\beta$ and $\alpha$

\[eq:pareto\] $$\begin{aligned}
    f^{\mathrm{Pareto}}(y_k;x,\beta,\alpha)=\left\{\begin{matrix}
    \alpha\beta^\alpha (y_k-x)^{-(\alpha+1)}&y_k\geq x+\beta, \\ 
    0&y_k< x+\beta. 
    \end{matrix}\right.
    \label{eq:pareto_pdf}
    \end{aligned}$$ and the CDF is given by $$\begin{aligned}
    F^{\mathrm{Pareto}}(y_k;x,\beta,\alpha)=1-\left(\frac{\beta}{y-x}\right)^\alpha.
    \label{eq:pareto_cdf}
    \end{aligned}$$

For the BLUE we get,

$$\begin{aligned}
    \hat{x}_{\mathrm{BLUE}}^{\mathrm{Pareto}}(y_{1:N},\beta,\alpha) &= \frac{1}{N}\sum_{k=1}^{N}y_k - \frac{\alpha\beta}{\alpha-1},&\quad \alpha>1
    \\
    {\mathrm{MSE}}(\hat{x}_{\mathrm{BLUE}}^{\mathrm{Pareto}}(y_{1:N},\beta,\alpha)) &=\frac{\alpha\beta^2}{N(\alpha-1)^2(\alpha-2)},&\quad \alpha>2
    \end{aligned}$$

The RBLS theorem cannot be used in case Pareto-distributed noises. We provide an unbiased 