x+\frac{\beta}{N}\\\\N(x+\beta)
    \end{bmatrix}.
    \label{eq:exponential_unknown_E_gen}
    \end{aligned}$$

Following the same line of reasoning as in Section \[subsec:unknown\_hyper\_parameter\_uniform\], the unbiased estimator is given by the transformation

$$\begin{aligned}
    g(T(y_{1:N}))=\begin{bmatrix}\frac{1}{N-1}\left(NT_1(y_{1:N})-\frac{1}{N}T_2(y_{1:N})\right)\\\\    \frac{1}{N-1}\left(T_2(y_{1:N})-NT_1(y_{1:N})\right)\end{bmatrix},
    \end{aligned}$$

that gives $$\begin{aligned}
    {\mathbb{E}}\left(g(T(y_{1:N}))\right) = \begin{bmatrix}x\\\beta  \end{bmatrix}.
    \end{aligned}$$

Finally, the MVU estimator when the hyper parameter $\beta$ is unknown, is given by

$$\begin{aligned}
    \hat{x}_{\mathrm{MVU}}^{\mathrm{Exp}}(y_{1:N}) &= \frac{N}{N-1}y^{(1)}_{1:N} - \frac{1}{N(N-1)} \sum_{k=1}^{N} y_k \nonumber\\&= \frac{N}{N-1}y^{(1)}_{1:N}- \frac{1}{N-1} \bar{y},
    \end{aligned}$$

where $\bar{y}$ is the sample mean. Assuming that $N$ is large $\min_k y_k$ and $\bar{y}$ are independent and the MSE of the estimator, asymptotically, is given by $$\begin{aligned}
    {\mathrm{MSE}}\left(\hat{x}_{\mathrm{MVU}}^{\mathrm{Exp}}(y_{1:N})\right) = \frac{\beta^2(N+1)}{N(N-1)^2}.
    \end{aligned}$$

Rayleigh distribution {#subsec:rayleigh_distribution}
---------------------

One generalization of the exponential distribution is obtained by parameterizing in terms of both a scale parameter $\beta$ and a shape parameter $\alpha$. Rayleigh distribution is a special case obtained by setting $\alpha=2$

\[eq:rayleigh\] $$\begin{aligned}
    f^{\mathrm{Rayleigh}}(y_k;x,\beta)=\left\{\begin{matrix}
    \frac{y_k-x}{\beta^2}\exp(-\frac{(y_k-x)^2}{2\beta^2})&y_k> x, \\ 
    0&y_k\leq x. 
    \end{matrix}\right.
    \label{eq:rayleigh_pdf}
    \end{aligned}$$ and the CDF, for $y_k>x$ is given by $$\begin{aligned}
    F^{\mathrm{Rayleigh}}(y_k;x,\beta)=1-\exp(-\frac{(y_k-x)^2}{2\beta^2}).
    \label{eq:rayleigh_cdf}
    \end{aligned}$$

Hence, the BLUE estimator , becomes

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

The marginal density of the $k$:th order statistic is given by $$\begin{aligned}
f^{\mathrm{Rayleigh}}_{(k,N)}(y;x,\beta) =\left\{\begin{matrix}
\frac{Ny}{\beta^2}\begin{pmatrix}N-1\\k-1\end{pmatrix}\left(1-\exp(-\frac{(y-x)^2}{2\beta^2})\right)^{k-1}\exp\left(-\frac{(N-k+1)(y-x)^2}{2\beta^2}\right)&y> x, \\ 
0&y\leq x. 
\end{matrix}\right. 
\label{eq:rayleigh_order}\end{aligned}$$ Hence, the minimum order statistics density also is Rayleigh 