### Article

## Сравнение искаженных гистограмм вероятностными методами

This paper is devoted to study of stability of comparison of histograms with help of different probability methods. Background: The comparison of histograms is necessary in many applied problems of data processing. The comparison of type ”more-less” is considered in this paper. But the histograms may be distorted. The nature of these distortions can be different. Then we have a problem to find the conditions on distortions under which the comparison of the two histograms is not changed. Methods: There are many approaches to comparison of histograms. The three popular proba- bilistic methods of comparison of histograms are considered in this paper: comparison of math- ematical expectations, comparison with help of principle of stochastic dominance, comparison with help of stochastic precedence. We consider the interval distortions of histograms in this paper. Results:The necessary and sufficient conditions of preservation for comparison of distorted his- tograms found with respect to different probability indices of comparison. The description of set of admissible distortions preserving the comparison of two histograms found. The characteristics of stability of histograms to distortion are introduced. These characteristics are calculated for histograms of USE (Unified State Exam) of applicants admitted in 2012 in Russian universities. It is shown that the stability of comparison of histograms to distortion can does not correspond to the values of difference index of comparison (margin). Conclusions: The found conditions invariability of comparing histograms can be used to es- timate the reliability of results of different rankings, data processing, etc. in terms of different types of uncertainty: stochastic uncertainty, the uncertainty associated with the distortion of the data in filling data gaps, etc.