On the Evaluation of Confidence Levels with Application to GNSS

The goal of this contribution is to assess the impact statistical model selection has on confidence levels of parameter estimators in linear(ized) GNSS models. In the processing of observational data, parameter estimation and statistical testing are often combined. A testing procedure is exercised to select the most likely observational model among the hypothesized ones, which is then followed by the estimation of the identified model parameters. The resulting estimator will inherit the uncertainties involved in both estimation and testing which need to be properly taken into account when computing the corresponding confidence level. The approach that is usually followed in practice to determine the confidence level is to compute the probability of the estimator lying in a region around its true value conditioned on the identified hypothesis. Therefore, use is made of the estimator’s distribution under the identified hypothesis without regard to the conditioning process that led to the decision of accepting this hypothesis. In this contribution, it will be shown that for a proper computation of the confidence level in combined estimation-testing procedures, the associated probability should be conditioned not only on the identified hypothesis, but also on the testing outcome that led to the decision of accepting this hypothesis. Therefore, use need to be made of the conditional distribution of the estimator. We will provide numerical analysis of confidence levels with and without accounting for conditioning on testing decision using a number of examples in the context of GNSS single point positioning. It will be demonstrated that the customary practice which makes use of unconditional distributions to evaluate the confidence level, may give a too optimistic description of the estimator’s quality.