Exploring Positive Noise in Estimation Theory

Estimation of the mean of a stochastic variable observed in noise with positive support is considered. It is well known from the literature that order statistics gives one order of magnitude lower estimation variance compared to the best linear unbiased estimator (BLUE). We provide a systematic survey of some common distributions with positive support, and provide derivations of minimum variance unbiased estimators (MVUE) based on order statistics, including BLUE for comparison. The estimators are derived with or without knowledge of the hyperparameters of the underlying noise distribution. Though the uniform, exponential and Rayleigh distributions, respectively, we consider are standard in literature, the problem of estimating the location parameter with additive noise from these distribution seems less studied, and we have not found any explicit expressions for BLUE and MVUE for these cases. In addition to additive noise with positive support, we also consider the mixture of uniform and normal noise distribution for which an order statistics-based unbiased estimator is derived. Finally, an iterative global navigation satellite system (GNSS) localization algorithm with uncertain pseudorange measurements is proposed which relies on the derived estimators for receiver clock bias estimation. Simulation data for GNSS time estimation and experimental GNSS data for joint clock bias and position estimation are used to evaluate the performance of the proposed methods.