parameters of the noise distributions are also derived, if possible. Finally, we derive an estimator for the case in which the noise follows a mixture of normal and uniform distribution. The rest of this paper is structured as follows. In Section \[sec:marginal\_distribution\_of\_order\_statistics\] the marginal distribution of order statistics is introduced. In Section \[sec:location\_estimation\_problem\] the location estimation problem is formulated. The problem is then investigated for different noise distributions and estimators for each distribution are derived in Sections \[sec:uniform\_distribution\]–\[sec:other\_distributions\]. The proposed estimators are evaluated in a simulation study in Section \[sec:performance\_evaluation\] followed by the concluding remarks given in Section \[sec:conclusions\].

Marginal Distribution of Order Statistics {#sec:marginal_distribution_of_order_statistics}
=========================================

The marginal distribution of order statistics, in this work is computed by differentiating the corresponding cumulative distribution function (CDF). In this section, we first introduce the minimum, also know as first or extreme, order statistic and then give the generalization to any statistics of order $k$. Let $F$ denote the common CDF of $N$ independent and identically distributed sample of random variables $y_1,\ldots,y_N$. We let $y^{(k)}$ denote the $k$:th order statistic of the sample, defined as the $k$:th smallest value of the set $\{y_i\}_{i=1}^N$. We define $f_{(k,N)}(y)$ as the marginal PDF of the $k$:th 