M (split(q)) estimation: estimation of parameters in a multi split functional model of geodetic observations

The method presented here assumes that a single observation can be identified with one of q functional models that compete with one another. The estimation method is based on the assumption that a theoretical quantity, called elementary split potential, can be assigned to each observation. Such quantity is referred to the theory of probability as well as to the theory of information. Parameters of the competitive functional models are estimated by maximizing the split potential globally over for the whole observation set. Additionally, such M (split(q)) estimates minimize the amount of information that could be provided by other estimates computed for the same observation set. The method is a certain kind of extension of the maximum likelihood method and if one considers the generalizations presented in the paper it can also be regarded as the development of M-estimation. Special attention is paid to the squared M (split(q)) estimation where the objective function is a squared one. If q = 1, then the squared M (split(q)) estimation is equivalent to the least squares method. The last part of the paper presents some numerical examples illustrating the properties of the squared M (split(q)) estimation as well as pointing at possible applications in geodesy and surveying.