Original article Forecasting trends in the development of time series by estimating the Hodges-Lehman median

Introduction. The lack of stationarity in the time series’ development, small samples, the outliers presence, jumps do not allow to find estimates of model parameters that have good properties of statistical estimates and, as a result, to find reliable forecasts, both in the development trend of the process and in its numerical expression. The means to solve these problems is the use of ordinal or robust statistics. Scientific monographs and special chapters in books on mathematical statistics contain a deep and extensive theory on the study of the properties of ordinal statistics, which are the justification for their application in forecasting methods. The aim of the work is to develop and verify method for obtaining one-step forecasts of trends in the development of time series based on stable statistical estimates.

Materials and methods. The article presents the results of the development of a method for obtaining one-step forecasts of trends in the development of time series based on the construction of confidence intervals of a selective stable HodgesLehman estimate based on Walsh averages. In particular, the Hodges-Lehman median is used to solve the problem of small samples obtained during the procedure of shifting the time series window. The proposed method is considered in detail in the article: the basic definitions, the theoretical justification of the method, calculation formulas, a detailed description of the algorithm, formulas for calculating the quality metric of forecasts are given.

The results of the study. The method was implemented in computational experiment using the example of forecasting the Urals crude oil’s spot price. The article presents the computational experiment’s results. The parameters of the proposed method can be configured to obtain reliable one-step forecasts.

Discussion and conclusions. The method proposed in the article has shown its effectiveness on experimental data and can be used as an independent method for constructing one-step forecasts of trends in the development of time series. Further development of the method involves the improvement of computational procedures, verification of the method in case of jumps in the dynamics of the time series.

1. Дейвид, Г. Порядковые статистики / Г. Дейвид. — Москва : Наука. Главная редакция физико-математической литературы, 1979. — 336 с.

2. Кобзарь, А. И. Прикладная математическая статистика. Для инженеров и научных работников / А. И. Кобзарь. — Москва : ФИЗМАТЛИТ, 2006. — 816 с.

3. Сархан, А. Е. Введение в теорию порядковых статистик / А. Е. Сархан, В. Гринберг / пер. с англ. под ред. Ф.Я. Боярского. — Москва : Статистика, 1970 г. — 414 с.

4. Мисюра, В. В. Прогнозирование тенденций развития случайных процессов с помощью порядковой статистики/ В. В. Мисюра // Инженерный журнал Дона. — 2018, № 4(51). — с. 1155.

5. Шуленин, В. П. Математическая статистика: Непараметрическая статистика / В. П. Шуленин / Ч. 2. — Томск : Изд-во НТЛ, 2012 г.

6. Hodjes, J. L. Estimation of location based on rank tests / J. L. Hodjes, E. L. Lehmann // The Annals of Mathematical Statistics. — 1963. — No. 4. — P. 598−611.