CDF of the error values used in the simulations are presented in Figure \[fig:cdf\_error\]. The support of the noise values, as can be read from the figure, is $\bm{e}_m\in[0,60]$ unit.

Let $\hat{x}_N^{(m)}$ denote the estimated value of the unknown parameter $x$ in the $m$:th repetition obtained from a sample of size $N$. For each noise distribution, the estimators’ performances are evaluated in terms of the obtained MSEs. The theoretical MSE of each estimator, as defined in Table \[tbl:estimators\] and Table \[tbl:estimators\_biased\], is compared against the numerical MSE obtained in simulations.

We let ${\mathbb{E}}[\hat{x}_N] = \frac{1}{M}\sum_{m=1}^{M}\hat{x}_N^{(m)}$ and define

$$\begin{aligned}
    \hat{b}_N &= {\mathbb{E}}[\hat{x}_N] - x\\
    \hat{\sigma}^2_N &= \frac{1}{M}\sum_{m=1}^{M}(\hat{x}_N^{(m)}-{\mathbb{E}}[\hat{x}_N])^2.
    \end{aligned}$$

The numerical MSE for each sample size $N$ is then computed by $$\begin{aligned}
    \hat{\mathrm{MSE}}(\hat{x}_N) = \hat{\sigma}^2_N + \hat{b}_N^2.
    \end{aligned}$$

![Analytical (marked with solid lines) and numerical (marked with crosses) MSE for uniform noise distribution as a function of the sample size $N$.[]{data-label="fig:uniform_blue_mvu"}](cdf_error.pdf){width="1.14\linewidth"}

![Analytical (marked with solid lines) and numerical (marked with crosses) MSE for uniform noise distribution as a function of the sample size $N$.[]{data-label="fig:uniform_blue_mvu"}](estimationMse_unif.pdf){width="\linewidth"}

Simulation Results
------------------

Figure \[fig:uniform\_blue\_mvu\] presents the performance of the four estimators when the noise is uniformly distributed. The solid lines correspond to the theoretical MSEs and the 