ОЦІНЮВАННЯ ЕКСПЕРИМЕНТАЛЬНОЇ ФУНКЦІЇ РОЗПОДІЛУ НА ОСНОВІ СКІНЧЕННИХ ВИБІРОК ВИПАДКОВОЇ ВЕЛИЧИНИ

• Дата: 2025
• Автори: Lozynsky , A.B., Romanyshyn , I.M., Rusyn , B.P.
• Формат: Стаття
• Мова: English
• Опубліковано: V.M. Glushkov Institute of Cybernetics of NAS of Ukraine 2025
• Теми: випадкова величина; функція розподілу; варіаційний ряд
• Онлайн доступ: https://jais.net.ua/index.php/files/article/view/692
• Назва журналу: Problems of Control and Informatics

A new approach to estimating the distribution function of a random variable based on finite (including small) samples is proposed. The approach is based on determining estimates of the positions of points of the distribution function. The theoretical substantiation of the approach and the algorithms for estimating the distribution function are presented. The equations (integral) for determining of the points on the desired distribution function are exact. Therefore, with knowledge of the distribution of any order statistics, the points of the sought-for distribution function are determined exactly. In the simplest case, the approach can be reduced to determining the median of statistics. Usually its exact value is unknown. Therefore, the estimation of median of the sample distribution is used as the median of the distribution of statistics. The consistency, bias and effectiveness of estimates of the points position on the sought-for distribution function are considered. It is shown that the proposed estimates are consistent, but biased, and the bias depends on the type of the sought-for distribution and decreases with increasing size of the subsample. Numerical experiments for the most important practical cases indicate an increasing of the effectiveness of the proposed approach compared to the classical one. Simplified algorithms for the approximate determination of distribution parameters are proposed, and error estimates are given that indicate the acceptability of the proposed simplified algorithms. The results of numerical modeling are presented, which illustrate the correspondence of the constructed distribution function of the model and the advantages of the approach. It is noted that the estimation of the distribution function based on the proposed approach in the form of an interpolated curve allows for numerical differentiation, filtering and other signal processing operations. At the same time, the distribution function in the classical form has a number of limitations in the application of these processing operations. A method for filtering a signal with uncorrelated pulse interference based on the proposed approach is described. The Appendix gives the determination of abscissas and ordinates of the sought-for distribution function of a random variable due to the solution of integral equations in an analytical form for the case of a uniform dis-tribution based on subsamples of two elements. The resulting solution is fully consistent with simplified algorithms.