and MSE of minimum order statistics estimators $\hat{x}_{\mathrm{min}}^{p_e}$.

\[tbl:estimators\_biased\]

\[tbl:estimators\]

Performance Evaluation {#sec:performance_evaluation}
======================

The estimators (both unbiased and the ones without bias compensation) derived in sections \[sec:distributioons\_in\_the\_exponential\_family\]–\[sec:other\_distributions\] for different noise distributions together with their MSE are summarized in Tables \[tbl:estimators\] and \[tbl:estimators\_biased\]. The biased minimum order statistics based estimators and their MSE are also The estimators derived for each noise distribution are compared against each other as a function of the sample size $N\in[2,\ldots,2000]$. Additionally, in order to verify the analytical derivations of the estimator variances, they are compared against the numerical variances obtained from $M=5000$ Monte Carlo runs.

Simulation Setup
----------------

For each sample size, $N$ noisy measurements of the unknown parameter $x$ are generated. The hyper parameters of the noise distributions are randomly selected in each repetition. In order to have a fair comparison, the hyper parameters are randomly drawn such that the error densities are mostly in the same range for all scenarios. The noise realizations are generated from the six considered distributions with the following hyper parameters

-   Uniform noise: $\beta\sim\mathcal{U}[6,50]$

-   Exponential noise: $\beta\sim\mathcal{U}[5,14]$

-   Rayleigh noise: $\beta\sim\mathcal{U}[5,12]$

-   Weibull noise:$\beta=1$, $\alpha\sim\mathcal{U}[5,10]$

-   Pareto noise: $\beta=6$, $\alpha\sim\mathcal{U}[2.1,2.5]$

-   Mixture noise: $\sigma\sim\mathcal{U}[1,9]$, $\beta\sim\mathcal{U}[1,50]$

The empirical 