Introduction. K.F. Gausse creates the classical least-square method (LSM) based on the hypothesis of normality of the observation errors. However this hypothesis, as a rule, is uncapable, if the number of the numerous instrumentation is n>500. In this case the errors are described with the Pearson-Jeffreys symmetrical threeparametrical distribution, which as the Gausse law has the diagonal information matrix and one can name it as the universal distribution law of overall size errors in accordance with numerous research. The aim of this investigation is elaboration of evolutionary procedures of the modern update LSM adapted to the Pearson-Jeffreys law of errors.

Methods of solving this problem is based on the analytic theory adapted to errors of observation weighting function that we have developed. The basic result of the work is that this theory transforms robust estimation from heuristic attempts into true science. The scientific novelty of this investigation: the meaning of the residual errors analysis was shown firstly from the Fisher’s theory of estimation point of view and it allows to outline the weight functions zones of singularity when LSM using. Practical importance: the diagnostic technique of the results of LSM usage on the basis of analysis of statistical semi-invariant of residual errors was elaborated and well-grounded evolutional procedures for receiving of effective LSM estimations which do not change practically the data handling classical algorithms were created.

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