Two-level probability distribution as a characteristic of the statistical stability of measurement object models

Consideration of the statistical stability of mathematical models of measurement objects, which significantly affects the accuracy of the statistical (probabilistic) estimates used, is considered. This problem is relevant because there is no effective criterion for distinguishing deterministic and random sequences. Accounting is based on a two-level probability distribution density in the form of a conversion formula. The numerator of the formula characterizes the errors of the systematic component of the hypothetical probability distribution as the observed deviations from it of the statistical distribution of measurement data, and the denominator is the unobservable components of the random component. The convergence of a number of repeated measurements is a necessary condition for the correctness of the statistical (probabilistic) estimates obtained. The formulation of such a problem is possible due to the fact that convolution in the form of an inversion formula is considered as the distribution of the sum of two random terms, when the second term characterizes the statistical stability of the first term. The direct solution of the problem by functional transformations of probability distributions leads to very cumbersome results. Within the framework of the interpolation concept, A. N. Kolmogorov's axiomatic approach is adopted, in which probability is represented by a positive real random variable characterized by a probability distribution function, which is a second-order distribution. It is established that the indicator of statistical stability of a mathematical model, the probability of agreement with the data of joint measurements, or kappa, is a reproducibility criterion, summarizes the statistics of the criteria of agreement of A. N. Kolmogorov and N. V. Smirnov – the distances between the distribution functions, and contains the distance according to the variation of V. Feller. The growth trend in the statistics of the probability of agreement with an increase in the sample size for probability distributions directly characterizes the degree of statistical stability of measurement data and the reliability of statistical inference logic in measuring problems of identifying probability distributions, and also complements confi dence probability as a characteristic of the quality of mathematical models of measurement objects.

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