Определение индекса Джини с учётом погрешностей выборочных наблюдений
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Дата
2019
DOI
doi.org/10.20998/2522-9052.2019.2.09
Науковий ступінь
Рівень дисертації
Шифр та назва спеціальності
Рада захисту
Установа захисту
Науковий керівник
Члени комітету
Видавець
Национальный технический университет "Харьковский политехнический институт"
Анотація
В статье рассматривается случайная величина, полученная в результате измерения величины конкретного физического свойства. Исследуются статистические и вычислительные аспекты задачи оценки отклонения распределения данной случайной величины от равномерного распределения. Рассматриваемая случайная величина нормируется. Одним из способов определения меры отклонения полученной таким образом функции распределения от равномерного распределения используют индекс Джини. Целью статьи является разработка предложений по учету погрешностей наблюдений при определении индекса Джини и построении кривой Лоренца. Результаты.Рассмотрена задача вычисления индекса Джини и параметров кривой Лоренца с учетом погрешностей выборочных наблюдений. Показано, что эта задача возникает в различных предметных областях, в том числе и социологии. Для
учёта ошибок выборочных наблюдений использованы интервальные вычисления в системе центр –радиус. Для вычисления индекса Джини применено численное интегрирование по формуле трапеций с использованием интервальных чисел. Вывод. Показано, что отказ от учёта ошибок выборочных наблюдений может привести к ошибочным выводам об уровне социального расслоения в обществе и оценке наличия в нём среднего класса.
In this work the following problem is considered. For a set of physically realizable objects, to each of them the value of its physically measurable property is assigned.The value of this property for a particular object is assumed to be random.The error in determining the numerical value of this property is negligible compared to the value of the property being measured. We assume that the value of the measured property is positive and finite. It is necessary to estimate thedeviation of the distribution of this property from the uniform distribution. Such problems arise in the study and management of processes in the various sy stems. For example, in concrete science, enrichment of the fields. Depending on the substantive meaning of the task, the goal of the research may be to find ways to get the maximum approximation of the density (function) of the property distribution to a uniform distribution or to achieve the opposite goal. The Gini index is chosen as a measure of the deviation of the distribution function of the measured property values from the uniform one, and the distribution itself is approximated by the Lorenz curve. The value of the Gini index is included in the system of indicators which characterizethe level of prosperity of the various states or units of territorial -administrative division. It is shown that statistical sampling methods are the main way of obtaining the necessary data for the construction of the Gini index. Was described the methodology of conducting sampling studies used in Poland. To perform thenecessary calculations in determining the Gini index and building a Lorenz curve, taking into account errors caused by using the sample data, the authors selected interval computing technology with the representation of numbers in the center–radius system. Тo calculate the Gini index, one of the methods of numerical integration is used - the trapezoid method. The implementation of this method is proposed in the interval form. The parameters of the expression approximating the Lorenz curve are determined.The calculation of the Gini index and the construction of the Lorenz curve is performed for information about weekly family income in the UK in 1992. It is shown that the refusal to take into account sampling errors can lead to erroneous conclusions when performing a comparative analysis of the uneven distribution of income between countries or their territories. A run test was made for checking the existence of a middle class in the surveyed population using two criteria. The first is based on comparing the Gini index value with the criterial index, the second one is based on comparing the difference between the third and first quartile of the sample described by the Lorenz curve with the criterial value. The results of numerical analysis for each of the criteria are given, which were performed using interval numbers determined in the center - radius system.
In this work the following problem is considered. For a set of physically realizable objects, to each of them the value of its physically measurable property is assigned.The value of this property for a particular object is assumed to be random.The error in determining the numerical value of this property is negligible compared to the value of the property being measured. We assume that the value of the measured property is positive and finite. It is necessary to estimate thedeviation of the distribution of this property from the uniform distribution. Such problems arise in the study and management of processes in the various sy stems. For example, in concrete science, enrichment of the fields. Depending on the substantive meaning of the task, the goal of the research may be to find ways to get the maximum approximation of the density (function) of the property distribution to a uniform distribution or to achieve the opposite goal. The Gini index is chosen as a measure of the deviation of the distribution function of the measured property values from the uniform one, and the distribution itself is approximated by the Lorenz curve. The value of the Gini index is included in the system of indicators which characterizethe level of prosperity of the various states or units of territorial -administrative division. It is shown that statistical sampling methods are the main way of obtaining the necessary data for the construction of the Gini index. Was described the methodology of conducting sampling studies used in Poland. To perform thenecessary calculations in determining the Gini index and building a Lorenz curve, taking into account errors caused by using the sample data, the authors selected interval computing technology with the representation of numbers in the center–radius system. Тo calculate the Gini index, one of the methods of numerical integration is used - the trapezoid method. The implementation of this method is proposed in the interval form. The parameters of the expression approximating the Lorenz curve are determined.The calculation of the Gini index and the construction of the Lorenz curve is performed for information about weekly family income in the UK in 1992. It is shown that the refusal to take into account sampling errors can lead to erroneous conclusions when performing a comparative analysis of the uneven distribution of income between countries or their territories. A run test was made for checking the existence of a middle class in the surveyed population using two criteria. The first is based on comparing the Gini index value with the criterial index, the second one is based on comparing the difference between the third and first quartile of the sample described by the Lorenz curve with the criterial value. The results of numerical analysis for each of the criteria are given, which were performed using interval numbers determined in the center - radius system.
Опис
Ключові слова
кривая Лоренца, интервальные вычисления, численное интегрирование интервально-определённой функции, Lorenz curve, interval calculations, numerical integration of an interval defined function
Бібліографічний опис
Дубницкий В. Ю. Определение индекса Джини с учётом погрешностей выборочных наблюдений / В. Ю. Дубницкий, Г. Г. Зубрицкая, А. И. Ходырев // Сучасні інформаційні системи = Advanced Information Systems. – 2019. – Т. 3, № 2. – С. 52-59.