estimator using minimum order statistics and its variance. The marginal density of the $k^{\mathrm{th}}$ order statistic, for $y\geq x+\beta$ is given by $$\begin{aligned}
f^{\mathrm{Pareto}}_{(k,N)}(y;x,\beta,\alpha) = N\alpha\beta^\alpha (y-x)^{-(\alpha+1)}\begin{pmatrix}N-1\\k-1\end{pmatrix}\left(1-(\frac{\beta}{y-x})^\alpha\right)^{k-1}\left(\frac{\beta}{y-x}\right){\alpha(N-k)}.
\label{eq:pareto_order}\end{aligned}$$ The minimum order statistic has the same form of distribution $$\begin{aligned}
f^{\mathrm{Pareto}}_{(1,N)}(y;\beta,\alpha) = f^{\mathrm{Pareto}}(y;\beta,\bar{\alpha}),\end{aligned}$$ where $\bar{\alpha}=N\alpha$. The MSE of the minimum order statistic estimator is $$\begin{aligned}
{\mathrm{MSE}}(\hat{x}_{\mathrm{min}}^{\mathrm{Pareto}}(y_{1:N})) =  \frac{N\alpha\beta^2}{N\alpha-2}\end{aligned}$$ The unbiased estimator is thus given by,

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

No unbiased estimator for unknown hyper parameter case could be found for Pareto distribution.

Mixture of Normal and Uniform Noise Distribution
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Suppose the error is distributed as $$e_k\sim \alpha\mathcal{N}(0,\sigma^2) + (1-\alpha) \mathcal{U}[0,\beta],$$ where $\alpha$ is the mixing probability of the mixture distribution. Define $f^{\mathrm{\mathcal{U},\mathcal{N}}}(y_k)$ as the probability density function of the considered mixture distribution given by $$\begin{aligned}
\label{eq:mixture_pdf}
f&^{\mathrm{\mathcal{U},\mathcal{N}}}(y_k;x,\alpha,\sigma^2,\beta) =\nonumber\\
&\begin{cases}
\begin{aligned}
& \frac{a}{\sqrt{2\pi\sigma^2}}\exp\left[-\frac{(y_k-x)^2}{2\sigma^2}\right] +\frac{1-a}{\beta}     \end{aligned} & \parbox[t]{4cm}{\raggedright $0\leq y_k-x\leq \beta$}\\
\frac{a}{\sqrt{2\pi\sigma^2}}\exp\left[-\frac{(y_k-x)^2}{2\sigma^2}\right] & \parbox[t]{4cm}{\raggedright Otherwise.}\\
\end{cases} \end{aligned}$$ The BLUE, in case of the mixture of normal and uniform measurement noises is given by

$$\begin{aligned}
    \hat{x}_{\mathrm{BLUE}}^{\mathrm{\mathrm{\mathcal{U},\mathcal{N}}}}(y_{1:N},\alpha,\beta,\sigma^2) &= \frac{1}{N}\sum_{k=1}^{N}y_k - \frac{\beta(1-\alpha)}{2},
    \\
    {\mathrm{MSE}}\left(\hat{x}_{\mathrm{BLUE}}^{\mathrm{\mathrm{\mathcal{U},\mathcal{N}}}}(y_{1:N},\alpha,\beta,\sigma^2)\right) &=\nonumber\\ &\frac{\beta^2  \left(1+(2-3\alpha)\alpha\right)+12\alpha\sigma^2}{12N}.
    \end{aligned}$$

Noting that at $y_k-x=0$ contributions of the uniform distribution and the mean (mode) of the normal distribution are 