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Statistical modeling is fundamentally based on probability distributions, which can be discrete or continuous and univariate or multivariate. This review focuses on the methods used to construct these distributions, covering both traditional and newly developed approaches. We first examine classic distributions such as the normal, exponential, gamma, and beta for univariate data, and the multivariate normal, elliptical, and Dirichlet for multidimensional data. We then address how, in recent decades, the demand for more flexible modeling tools has led to the creation of complex meta-distributions built using copula theory.

We report a new population of outer belt electron acceleration events ranging from hundreds of keV to ∼1.5 MeV that occurred inside the plasmasphere, which we named “Inside Events” (IEs). Based on 6 year observations from Van Allen Probes, we compare the statistical distributions of IEs with electron acceleration events outside the plasmasphere (OEs). We find that most IEs were observed at L < 4.0 at energies below ∼1.5 MeV, with weaker acceleration ratio (<10) and larger event numbers (peaking value reaching >200), compared to stronger but less frequently occurred (peaking event numbers only reaching ∼80) OEs that were mostly observed at L > 4.0. The evolution of electron phase space density of a typical IE shows signature of inward radial diffusion or transport. Our study provides a feasible mechanism for IE, which is the results of the inward radial transport of the electron acceleration in the outer region of outer belt. Plain Language Summary Since the discovery of the Earth's Van Allen radiation belts in 1958, extensive studies have advanced our understanding of outer belt electron acceleration mechanisms. However, most previous studies focused on the electron acceleration process occurring in the low electron density region, outside the plasmasphere. Although limited previous studies reported the electron flux enhancements penetrated down to very low L‐shells (L = 2.5), involving flux enhancements inside the plasmasphere, these studies did not specify the plasmapause location. In this letter, we report a new population of outer belt electron acceleration events ranging from hundreds of keV to ∼1.5 MeV observed inside the plasmasphere. The “inside electron acceleration events” (IEs) are weaker but occur much more frequently compared to the stronger acceleration events observed outside plasmasphere, and cannot be neglected when investigating radiation belt electron dynamics. The evolution of electron phase space density (PSD) of a typical IE event demonstrates signature of inward radial transport, showing gradual flux enhancements over several days and monotonically increasing radial profile of electron PSD. Our study provides convincing evidence that this observed IE in the low L‐shell region (L = 2.5) was dominantly caused by inward radial transport of electron acceleration in the outer region of the outer belt. Key Points We report a new population of outer belt electron acceleration events from 300 keV to ∼1.5 MeV observed inside the plasmasphere (IE) At >300 keV, the IEs are weaker but occur more frequently compared to the stronger acceleration events observed outside the plasmasphere The evolution of electron phase space density of a typical IE shows monotonically increasing radial profile, suggesting the crucial role of radial transport

This paper presents a new methodology for generating continuous statistical distributions, integrating the exponentiated odds ratio within the framework of survival analysis. This new method enhances the flexibility and adaptability of distribution models to effectively address the complexities inherent in contemporary datasets. The core of this advancement is illustrated by introducing a particular subfamily, the “Type 2 Gumbel Weibull-G family of distributions”. We provide a comprehensive analysis of the mathematical properties of these distributions, including statistical properties such as density functions, moments, hazard rate and quantile functions, Rényi entropy, order statistics, and the concept of stochastic ordering. To test the robustness of our new model, we apply five distinct methods for parameter estimation. The practical applicability of the Type 2 Gumbel Weibull-G distributions is further supported through the analysis of three real-world datasets. These real-life applications illustrate the exceptional statistical precision of our distributions compared to existing models, thereby reinforcing their significant value in both theoretical and practical statistical applications.

Statistical modeling relies on a diverse range of statistical distributions, encompassing both univariate and multivariate distributions and/or discrete and continuous distributions. In the literature, numerous statistical methods have been proposed to approximate continuous distributions. The most commonly used approach is the use of the empirical distribution which is obtained from a random sample drawn from the distribution. However, it is very likely that the empirical distribution suffers from an accuracy problem when used to approximate the underlying distribution, especially if the sample size is not sufficient. In order to improve statistical inferences, various alternative forms of discrete approximation to the distribution were proposed in the literature. The choice of support points for the discrete approximation, known as Representative Points (RPs), becomes extremely important in terms of distribution approximations. In this paper we give a review of the three main methods for constructing RPs, namely based on the Monte Carlo method, the number-theoretic method (or quasi-Monte Carlo method), and the mean square error method, aiming to introduce such important methods to the statistical or mathematical community. Additional approaches for forming RPs are also briefly discussed. The review focuses on certain critical aspects such as theoretical properties and computational algorithms for constructing RPs. We also address the issue of the application of RPs through studying practical problems and provide evidence of RPs’ advantages over random samples in approximating the distribution.

Research into statistical distributions of φ, λ and two-dimensional (2D) position errors of the global positioning system (GPS) enables the evaluation of its accuracy. Based on this, the navigation applications in which the positioning system can be used are determined. However, studies of GPS accuracy indicate that the empirical φ and λ errors deviate from the typical normal distribution, significantly affecting the statistical distribution of 2D position errors. Therefore, determining the actual statistical distributions of position errors (1D and 2D) is decisive for the precision of calculating the actual accuracy of the GPS system. In this paper, based on two measurement sessions (900,000 and 237,000 fixes), the distributions of GPS position error statistics in both 1D and 2D space are analysed. Statistical distribution measures are determined using statistical tests, the hypothesis on the normal distribution of φ and λ errors is verified, and the consistency of GPS position errors with commonly used statistical distributions is assessed together with finding the best fit. Research has shown that φ and λ errors for the GPS system are normally distributed. It is proven that φ and λ errors are more concentrated around the central value than in a typical normal distribution (positive kurtosis) with a low value of asymmetry. Moreover, φ errors are clearly more concentrated than λ errors. This results in larger standard deviation values for φ errors than λ errors. The differences in both values were 25–39%. Regarding the 2D position error, it should be noted that the value of twice the distance root mean square (2DRMS) is about 10–14% greater than the value of R95. In addition, studies show that statistical distributions such as beta, gamma, lognormal and Weibull are the best fit for 2D position errors in the GPS system.

The most used approach to solve tolerance analysis problems for flexible assemblies is the method of influence coefficients that combines the finite element analysis with statistical analysis in order to establish a relationship between the assembly deviation and part deviation and to foresee the statistical distribution of stresses. The key of this relationship is the sensitivity matrices for the deviations and stresses that can be evaluated by different methods of influence coefficients. Therefore, the aim of this work is to make a review of these methods applying them to evaluate some flexible assemblies on the statistical distribution of deviations and stresses.

The tensile properties and the Weibull statistical distributions of the tensile strength of poly-(para-phenylene-2,6-benzobisoxazole) (PBO), poly-(para-phenylene terephthalamide) (PPTA), copoly-(para-phenylene-3,4′-oxydiphenylene terephthalamide (PPODTA), polyarylate (PAR), and polyethylene (PE) polymeric fiber epoxy-impregnated bundle composites have been investigated. The results show that the Weibull modulus decreases as the tensile modulus, strength, and inverse of the failure strain increase. The interfacial shear properties were also examined using the microdroplet composite. For the lower interfacial shear strength of polymeric fibers, the Weibull modulus decreases as interfacial shear strength increases. Conversely, for the higher interfacial shear strength of polymeric fibers, the Weibull modulus increases as interfacial shear strength increases. Interestingly, these inflection points were also observed for the 20–30 MPa interfacial shear strength.

This review presents the main results of the author’s study, obtained as part of the post-doctoral (habilitation) dissertation entitled “Research on Statistical Distributions of Navigation Positioning System Errors”, which constitutes a series of five thematically linked scientific publications. The main scientific aim of this series is to answer the question of what statistical distributions follow the position errors of navigation systems, such as Differential Global Positioning System (DGPS), European Geostationary Navigation Overlay Service (EGNOS), Global Positioning System (GPS), and others. All of the positioning systems under study (Decca Navigator, DGPS, EGNOS, and GPS) are characterised by the Position Random Walk (PRW), which means that latitude and longitude errors do not appear randomly, being a feature of the normal distribution. The research showed that the Gaussian distribution is not an optimal distribution for the modelling of navigation positioning system errors. A higher fit to the 1D and 2D position errors was exhibited by such distributions as beta, gamma, and lognormal. Moreover, it was proven that the Twice the Distance Root Mean Square (2DRMS(2D)) measure, which assumes a priori normal distribution of position errors in relation to latitude and latitude, was smaller by 10–14% than the position error value from which 95% fixes were smaller (it is known as the R95(2D) measure).

In this paper, some statistical properties of the Choquet integral are discussed. As an interesting application of Choquet integral and fuzzy measures, we introduce a new class of exponential-like distributions related to monotone set functions, called , by combining the properties of Choquet integral with the exponential distribution. We show some famous statistical distributions such as gamma, logistic, exponential, Rayleigh and other distributions are a special class of Choquet distributions. Then, we show that this new proposed Choquet exponential distribution is better on daily gold price data analysis. Also, a real dataset of the daily number of new infected people to coronavirus in the USA in the period of 2020/02/29 to 2020/10/19 is analyzed. The method presented in this article opens a new horizon for future research.

What do we mean by inequality comparisons? If the rich just get richer and the poor get poorer, the answer might seem easy. But what if the income distribution changes in a complicated way? Can we use mathematical or statistical techniques to simplify the comparison problem in a way that has economic meaning? What does it mean to measure inequality? Is it similar to National Income? Or price index? Is it enough just to work out the Gini coefficient? This book tackles these questions and examines the underlying principles of inequality measurement and its relation to welfare economics, distributional analysis, and information theory. The book covers modern theoretical developments in inequality analysis, as well as showing how the way we think about inequality today has been shaped by classic contributions in economics and related disciplines. Formal results and detailed literature discussion are provided in two appendices. The principal points are illustrated in the main text, using examples from US and UK data, as well as other data sources, and associated web materials provide hands-on learning.