Impact of Outlier Monitoring on Confidence Regions: GNSS Positioning Examples

This contribution aims to evaluate how outlier monitoring impacts the confidence regions of parameter estimators in Global Navigation Satellite System (GNSS) linearized models. In the process of data collection from GNSS satellites, the presence of outliers is inevitable. Therefore, outlier monitoring mechanisms are employed when deriving estimates of parameters of interest from GNSS data. These mechanisms often involve a statistical testing procedure in which an outlier-free null hypothesis is compared against multiple outlier-describing alternative hypotheses. The outcome of this testing dictates how the parameters of interest, e.g., receiver position, are estimated. As a result, the characteristics of the obtained parameter estimator are influenced by both the estimation and testing steps. This estimator does not follow a normal distribution due to the impact of testing on estimation. However, the customary procedure in practice does not account for the uncertainty of the testing decisions and uses normal distributions, leading to an incorrect description of the estimator’s quality, such as faulty confidence regions. In this contribution, we explore the interaction between estimation and testing, demonstrating how their combined non-normal distribution can be used to create accurate confidence regions. Our focus is on the pre-measurement design phase, operating under the assumption that the null hypothesis is true. We compare three different confidence regions as follows: (I) An ellipsoidal confidence region neglecting the impact of testing preceding the estimation process, with its size determined by a normal distribution; (II) An ellipsoidal confidence region with its size determined by the actual Probability Density Function (PDF) of the parameter estimator; (III) A confidence region with both shape and size determined by the contours of the actual PDF of the parameter estimator. Using a simple example, the differences between these confidence regions are illustrated. Our numerical analysis is then continued within the framework of GNSS-based positioning. It is shown that confidence region (I), commonly used in practice, has poor coverage, yielding overly optimistic results and necessitating an expansion to encompass the desired probability for the parameter estimator.