distributed $$\begin{aligned}
f^{\mathrm{Rayleigh}}_{(1,N)}(y;x,\beta) = f^{\mathrm{Rayleigh}}(y;x,\bar{\beta}),\end{aligned}$$ where $\bar{\beta}=\frac{\beta}{\sqrt{N}}$. The MSE of the minimum order statistics is given by $$\begin{aligned}
{\mathrm{MSE}}\left(\hat{x}^{\mathrm{Rayleigh}}_{\mathrm{min}}(y_{1:N})\right) = \frac{2\beta^2}{N}.\end{aligned}$$ The joint PDF of $N$ independent observations $y_{1:N}$ is given by

\[eq:rayleigh\_pdf\_2\] $$\begin{aligned}
    f(y_{1:N};x,\beta) &= \frac{\prod_{k=1}^{N}(y_k-x)}{\beta^{2N}}\exp\left[\sum_{k=1}^{N}-\frac{(y_k-x)^2}{2\beta^2}\right]\sigma(y^{(1)}_{1:N}- x).
    \label{eq:rayleigh_pdf_2_1}
    \end{aligned}$$ Noting that $$\begin{aligned}
    \sum_{k=1}^{N}(y_k-x)^2 = \sum_{k+1}^{N}(y_k)^2 - 2x\sum_{k=1}^{N}y_k+Nx^2,
    \label{eq:rayleigh_pdf_2_2}
    \end{aligned}$$ the PDF becomes $$\begin{aligned}
    f(y_{1:N};x,\beta) &= \beta^{-2N}\prod_{k=1}^{N}(y_k-x)\exp\left[\frac{-1}{2\beta^2}\sum_{k=1}^{N}y_k^2\right]\nonumber\\&\times\exp\left[-\frac{Nx^2}{2\beta^2}\right]\exp\left[\frac{x}{\beta^2}\sum_{k=1}^{N}y_k\right]\sigma(y^{(1)}_{1:N} - x).
    \label{eq:rayleigh_pdf_2_3}
    \end{aligned}$$

### Known hyper parameter $\beta$ {#known-hyper-parameter-beta-1}

Since  cannot be factorized in the form of $f(y_{1:N};x,\beta) = g(T(y_{1:N}),x)h(y_{1:N})$, the RBLS theorem cannot be used. Hence, even if an MVU estimator exists for this problem, we may not be able to find it. Thus, in case of Rayleigh-distributed measurement noise, we propose unbiased estimators based on order statistics.

If the hyper parameter of the distribution is known, the unbiased order statistic based estimator $\hat{x}^{\mathrm{Rayleigh}}(y_{1:N},\beta)$ is then given by,

$$\begin{aligned}
    \hat{x}^{\mathrm{Rayleigh}}(y_{1:N},\beta) &= y^{(1)}_{1:N} - \frac{\sqrt{\pi}\beta}{\sqrt{2N}},
    \\ 
    {\mathrm{MSE}}\left(\hat{x}^{\mathrm{Rayleigh}}(y_{1:N},\beta)\right) &=\frac{(4-\pi)\beta^2}{2N}.
    \end{aligned}$$

which has the same variance as the BLUE estimator.

### Unknown hyper parameter $\beta$ {#unknown-hyper-parameter-beta-1}

In case of unknown hyper parameters, as for the known case, no factorization that enables us to use the RBLS theorem can be found. In this case, we propose the following unbiased estimator $$\begin{aligned}
\hat{x}^{\mathrm{Rayleigh}}(y_{1:N}) &= \frac{\sqrt{N}}{\sqrt{N}-1}y^{(1)}_{1:N} - 