The purpose of this paper is to describe a statistical method for modelling and accounting for the heavy tail fault-free error distributions that have been encountered in the Local Area Augmentation System (LAAS), the FAA's version of a ground-based augmentation system (GBAS) for GPS. The method uses the Normal Inverse Gaussian (NIG) family of distributions to describe a heaviest tail distribution, and to select a suitable NIG family member as a model distribution based upon a statistical observability criterion applied to the FAA's LAAS prototype error data. Since the independent sample size of the data is limited to several thousand and the tail probability of interest is of the order of 10−9, there is a chance of mismodelling. A position domain monitor (PDM) is shown to provide significant mitigation of mismodelling, even for the heaviest tail that could be encountered, if it can meet certain stringent accuracy and threshold requirements. Aside from its application to GBAS, this paper should be of general interest because it describes a different approach to navigation error modelling and introduces the application of the NIG distribution to navigation error analysis.