AN ALGORITHM OF TREATMENT OF THE RESULTS OF MULTIPLE MEASUREMENTS OF COMPOSITION AND PROPERTIES OF PETROLEUM PRODUCTS

Разработаны новые робастные алгоритмы обработки результатов многократных измерений состава и свойств нефтепродуктов, учитывающие тот факт, что эмпирическая функция распределения результатов измерений состава и свойств нефтепродуктов представляет собой смесь двух нормальных распределений с разными значениями параметров положения и масштаба. В случае измерений состава и свойств нефтепродуктов в качестве робастных оценок параметра положения и параметра масштаба выборки предложено использовать М-оценки с предварительным масштабированием на основе модифицированной функции Хампеля. Для нахождения М-оценки предложены два итеративных способа вычисления на основе средневзвешенного метода наименьших квадратов, отличающиеся процедурами расчета начальных оценок параметров положения и масштаба выборки. При числе результатов в выборке более двадцати в качестве начальных значений параметров положения и масштаба целесообразно использовать α‑урезанное среднее и α‑урезанное стандартное отклонение с долей усечения 0,05. При числе результатов в выборке менее двадцати в качестве начальных значений параметра положения и параметра масштаба обоснованно использование робастных оценок, не требующих удаления части данных. В качестве начальной оценки параметра положения предложено использовать оценку Ходжеса – Лемана; в качестве параметра масштаба – медианы абсолютных разностей. Предложенные робастные алгоритмы могут быть использованы при обработке результатов эксперимента по определению показателей прецизионности, правильности и точности методик измерений состава и свойств нефтепродуктов, итогов межлабораторных сравнительных испытаний нефтепродуктов, расчете аттестованного значения стандартных образцов состава и свойств нефтепродуктов, а также в других случаях многократных наблюдений. New robust algorithms of treatment of the results of multiple measurements of composition and properties of petroleum products were developed in respect that empirical distribution function of the results of measurements of composition and properties of petroleum products are the mixture of two normal distributions with different values of position and scale parameters. In case of measurements of composition and properties of petroleum products it has been proposed to use M-estimator with pre-scaling based on modified Hampel function as robust estimators of position and scale parameters. To calculation M-estimator two iterative methods based on weighted average method of least squares were suggested which differs by procedures of initial estimators of position and scale parameters of sample. In case of more than twenty results in sample, it is expedient to apply α-truncated mean and α-truncated standard deviation with 0,05 truncation share as initial values of position and scale parameters. In case of less than twenty results in sample, it is reasonable to apply robust estimators as initial values of position and scale parameters, which don’t require removal of some part of the data. It was proposed to use Hodges-Lehmann estimator as an initial value of position parameter and median of absolute differences as a scale parameter. The proposed robust algorithms can be used in treatment of experiment results on determination of indexes of precision, trueness and accuracy of the methods of measurement of composition and properties of petroleum products; results of interlaboratory comparison tests of petroleum products; calculation of certified value of standard samples of composition and properties of petroleum products and in other cases of multiple observations.

Comparison of Robust Estimators’ Performance for Detecting Outliers in Multivariate Data

In multivariate data, outliers are difficult to detect especially when the dimension of the data increase. Mahalanobis distance (MD) has been one of the classical methods to detect outliers for multivariate data. However, the classical mean and covariance matrix in MD suffered from masking and swamping effects if the data contain outliers. Due to this problem, many studies used a robust estimator instead of the classical estimator of mean and covariance matrix. In this study, the performance of five robust estimators namely Fast Minimum Covariance Determinant (FMCD), Minimum Vector Variance (MVV), Covariance Matrix Equality (CME), Index Set Equality (ISE), and Test on Covariance (TOC) are investigated and compared. FMCD has been widely used and is known as among the best robust estimator. However, there are certain conditions that FMCD still lacks. MVV, CME, ISE and TOC are innovative of FMCD. These four robust estimators improve the last step of the FMCD algorithm. Hence, the objective of this study is to observe the performance of these five estimator to detect outliers in multivariate data particularly TOC as TOC is the latest robust estimator. Simulation studies are conducted for two outlier scenarios with various conditions. There are three performance measures, which are pout, pmask and pswamp used to measure the performance of the robust estimators. It is found that the TOC gives better performance in pswamp for most conditions. TOC gives better results for pout and pmask for certain conditions.

Comparison of robust logistic regression estimators for variables with generalized extreme value distributions

The aim of this study is to compare the performance of robust estimators in the presence of explanatory variables with Generalized Extreme Value (GEV) distributions in the logistic regression model. Existence of extreme values in the logistic regression model negatively affects the bias and effectiveness of classical Maximum Likelihood (ML) estimators. For this reason, robust estimators that are less sensitive to extreme values have been developed. Random variables with extreme values may be fit in one of specific distributions. In study, the GEV distribution family was examined and five robust estimators were compared for the Fréchet, Gumbel and Weibull distributions. To the simulation results, the CUBIF estimator is prominent according to both bias and efficiency criteria for small samples. In medium and large samples, while the MALLOWS estimator has the minimum bias, the CUBIF estimator has the best efficiency. The same results apply for different contamination ratios and different scale parameter values of the distributions. Simulation findings were supported by a meteorological real data application.

Robust Liu-Type Estimator for SUR Model

The Liu-type estimator is one of the shrink estimators that is used to remedy for a problem of multicollinearityin SUR model, but it is sensitive to the outlier. In this paper, we introduce the S Liu-type (SLiu-type) and MM Liu-type estimator (MMLiu-type) for SUR model. These estimators merge Liu-type estimator with S-estimator and with MM-estimator which makes it have high robustness at the different level of efficiency and at the same time prevents the bad effects of multicollinearity. Moreover, to get more robust features, we have modified the Liu-type estimator by making it depend on MM estimator instead of GLS estimator. The asymptotical properties for the suggested estimator were discussed and we used the fast and robust bootstrap (FRB) to obtain the suggested robust estimators. Furthermore, we run the simulation study to show the extent of excellence for the suggested robust estimators relative to the other estimators by many factors.

Global Optimization of Redescending Robust Estimators

Robust estimation has proved to be a valuable alternative to the least squares estimator for the cases where the dataset is contaminated with outliers. Many robust estimators have been designed to be minimally affected by the outlying observations and produce a good fit for the majority of the data. Among them, the redescending estimators have demonstrated the best estimation capabilities. It is little known, however, that the success of a robust estimation method depends not only on the robust estimator used but also on the way the estimator is computed. In the present paper, we show that for complicated cases, the predominant method of computing the robust estimator by means of an iteratively reweighted least squares scheme may result in a local optimum of significantly lower quality than the global optimum attainable by means of a global optimization method. Further, the sequential use of the proposed global robust estimation proves to successfully solve the problem of M-split estimation, that is, the determination of parameters of different functional models implicit in the data.