<p>The article examines the estimation of traditional descriptive statistics of the symmetric and asymmetric multimodal distributions of one-dimensional random variables for large amounts of statistical data. The authors comparatively analyze the effectiveness of methods for estimating these descriptive statistics when using the original statistical data and the results of data decomposition obtained by applying four formulas for random-variable discretization—Sturges’, Brooks-Carruthers, and Heinhold-Gaede formulas, as well as the optimal discretization formula proposed in this article. The use of these discretization formulas circumvents the problem associated with large samples. To this end, data arrays were formed, which provided a&#xa0;means to estimate the descriptive statistics of the probability distributions of random variables, taking their discrete values into account. The transformed data arrays were used to estimate the expected value and standard deviation, as well as skewness and kurtosis. The estimated descriptive statistics of the considered probability distributions of continuous and discrete random variables were compared for different amounts of the original statistical data. The effectiveness of methods for estimating the descriptive statistics of multimodal distributions was established for the original statistical data and the results of their transformation with the specified discretization formulas. The comparison validity was confirmed using the Kolmogorov–Smirnov test. The Heinhold-Gaede formula and the optimal discretization formula proposed by the present authors were shown to be more effective than Sturges’ and Brooks-Carruthers formulas. The obtained results can be used in processing remote sensing data of natural objects, which are characterized by a&#xa0;large amount of statistical data and multimodal probability distributions of spectral features.</p>

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Multimodal distributions of a one-dimensional random variable: Estimation of traditional descriptive statistics for large amounts of statistical data

  • Aleksandr V. Lapko,
  • Vasiliy A. Lapko

摘要

The article examines the estimation of traditional descriptive statistics of the symmetric and asymmetric multimodal distributions of one-dimensional random variables for large amounts of statistical data. The authors comparatively analyze the effectiveness of methods for estimating these descriptive statistics when using the original statistical data and the results of data decomposition obtained by applying four formulas for random-variable discretization—Sturges’, Brooks-Carruthers, and Heinhold-Gaede formulas, as well as the optimal discretization formula proposed in this article. The use of these discretization formulas circumvents the problem associated with large samples. To this end, data arrays were formed, which provided a means to estimate the descriptive statistics of the probability distributions of random variables, taking their discrete values into account. The transformed data arrays were used to estimate the expected value and standard deviation, as well as skewness and kurtosis. The estimated descriptive statistics of the considered probability distributions of continuous and discrete random variables were compared for different amounts of the original statistical data. The effectiveness of methods for estimating the descriptive statistics of multimodal distributions was established for the original statistical data and the results of their transformation with the specified discretization formulas. The comparison validity was confirmed using the Kolmogorov–Smirnov test. The Heinhold-Gaede formula and the optimal discretization formula proposed by the present authors were shown to be more effective than Sturges’ and Brooks-Carruthers formulas. The obtained results can be used in processing remote sensing data of natural objects, which are characterized by a large amount of statistical data and multimodal probability distributions of spectral features.