Estimating Noisy Order Statistics

This paper proposes an estimation framework to assess the performance of sorting over perturbed/noisy data. In particular, the recovering accuracy is measured in terms of Minimum Mean Square Error (MMSE) between the values of the sorting function computed on data without perturbation and the estimator that operates on the sorted noisy data. It is first shown that, under certain symmetry conditions, satisfied for example by the practically relevant Gaussian noise perturbation, the optimal estimator can be expressed as a linear combination of estimators on the unsorted data. Then, two suboptimal estimators are proposed and performance guarantees on them are derived with respect to the optimal estimator. Finally, some surprising properties on the MMSE of interest are discovered. For instance, it is shown that the MMSE grows sublinearly with the data size, and that commonly used MMSE lower bounds such as the Bayesian Cram\'er-Rao and the maximum entropy bounds either cannot be applied or are not suitable.