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The exponential family of probability distributions, in their most general form, is defined by $$\begin{aligned}
f(y;\theta) = h(y)g(\theta)\exp\left\{A(\theta)\cdot T(y)\right\},\end{aligned}$$ where $\theta$ is the parameter of the distribution, and $h(y)$, $g(\theta)$, $A(\theta)$, and $T(y)$ are all known functions. In this section, we only consider some example distributions of this family and show that the minimum order statistic estimator gets the same form of distribution as the noise distribution but with modified parameters. For the selected distributions, if possible, MVU estimators for both cases of known and unknown hyperparameter are derived. Otherwise, unbiased estimators with less variance than BLUE are proposed.

Exponential distribution {#subsec:exponential_distribution}
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Exponential distributions are members of the gamma family with shape parameter 1; strongly skewed with no left sided tail ($y_k\in[x,\infty]$). Let $\beta>0$ denote the scale parameter, the PDF of an exponential distribution is then given by

\[eq:exponential\] $$\begin{aligned}
    f^{\mathrm{Exp}}(y_k;x,\beta)=\left\{\begin{matrix}
    \frac{1}{\beta}\exp(-\frac{(y_k-x)}{\beta})&y_k\geq x, \\ 
    0&y_k< x. 
    \end{matrix}\right.
    \label{eq:exponential_pdf}
    \end{aligned}$$ and the CDF, for $y\geq x$, is given by $$\begin{aligned}
        F^{\mathrm{Exp}}(y_k;x,\beta)=1-\exp(-\frac{(y_k-x)}{\beta}).
        \label{eq:exponential_cdf}
        \end{aligned}$$

For the BLUE estimator , from the properties of exponential distribution, we have $$\begin{aligned}
\hat{x}_{\mathrm{BLUE}}^{\mathrm{Exp}}(y_{1:N},\beta) &= \frac{1}{N}\sum_{k=1}^{N}y_k - \beta,
&\quad 
{\mathrm{MSE}}(\hat{x}_{\mathrm{BLUE}}^{\mathrm{Exp}}) &= \frac{\beta^2}{N}.\end{aligned}$$ Substituting  into , the marginal density of the $k$:th order statistic is given 