The statistical testing of regularized mathematical models in geodetic data processing

Abstract

The geodetic community commonly challenges the composite hypotheses in the statistical testing of mathematical models. Since the composite hypotheses are not specified as opposed to their simple counterparts, they require a prior estimation of the model parameters. However, if the mathematical models are ill-conditioned, the regularized estimation is often applied for the parameters of interest. Due to the biased property, the regularized estimation does not rigorously originate in the principle of maximum likelihood (ML) estimation, which was the base for developing the theory of the generalized likelihood ratio (GLR) test. Since the regularized estimator of the parameters of interest is consequently inconsistent with the ML one, one cannot construct the GLR test, which is the uniformly most powerful invariant (UMPI) test. So far, only the bias correction approach has been suggested to solve this problem. In this contribution, an implicit representation of the regularized mathematical model is proposed. It eliminates the complete impact of regularized estimation on a mathematical model and delivers the misclosures analytically free from the influence of regularization. Thus, one can construct the GLR test, which belongs to the UMPI family, and then formulate the test statistic in terms of misclosures.