On Estimation under Noisy Order Statistics

This paper presents an estimation framework to assess the performance of the sorting function over data that is perturbed. In particular, the performance is measured in terms of the Minimum Mean Square Error (MMSE) between the values of the sorting function computed on the data without perturbation and the estimate that uses the sorting function applied to the perturbed data. It is first shown that, under certain conditions satisfied by the practically relevant Gaussian noise perturbation, the optimal estimator can be expressed as a linear combination of estimators on the unsorted data. Then, a suboptimal estimator is proposed, and its performance is evaluated and compared to the optimal estimator. Finally, a lower bound on the desired MMSE is derived when data is i.i.d. and has a Gaussian distribution. This is accomplished by solving a new problem that consists of estimating the norm of an unsorted vector from a noisy observation of it.