Solution to some statistical problems with applications on GNSS error distributions

In order to support a wide range of user applications demanding provision of high accuracy positioning along with the continuous availability of a reliable signal, integrity has been identified as one of the key concepts in the design of GNSS missions. Integrity performances are defined in association to confidence levels on the determination of several accuracy related parameters, representing the probability that the system fails in such a way that provides misleading information. The stringent and diverse demands in terms of Confidence Levels posed by different types of users (aeronautical, maritime, etc) can be taken into account by means of a parametric statistical distribution that bounds the error in the information provided by the system, which is in turn made up from several individual contributors. For convenience and simplicity, several different ways of combining gaussian distributions (with mean and variance parameters as solve-for values) are generally considered for the over-bounding of the cumulative probability of the experimental distribution (CDF-overbounding). This paper describes several solutions to statistical related problems (e.g. explicit formulas for the correlations between different statistical parameters and for the CDF-overbounding of the absolute value of error distributions, when the underlying experimental distribution is known) and the applications to some related statistical new developments. In practical terms this mathematical formulations allows for sensitivity analysis as a function of the variables involved in the estimation of the error bounds along with a sounded approach when dealing with experimental data and a limited number of samples (i.e. long-term characterization of distributions and achievable confidence levels for system operations). As an additional result, experimental error distribution can be classified, for specific definitions of the CDF-overbounding, within just a few categories of distributions, which can then be assessed in a similar way, greatly simplifying the mathematical analysis of the problem.