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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.

In laboratory fracture toughness studies, the crack growth resistance of rock materials may be influenced by different factors such as specimen geometry, loading conditions, and also the type of pre-notch cut in the test sample. In this paper, a large number of mode I fracture toughness experiments are conducted on an Iranian white rock “Harsin marble” with six different mode I specimens. The selected test specimens are in the shape of cylindrical rod, rectangular beam, and circular Brazilian disk containing either chevron notch or straight crack. The effect of specimen geometry and pre-notch type was investigated statistically, and it was found that the average fracture toughness values of notched specimens were higher than those of the similar specimens but containing straight crack. Meanwhile, the scatters of fracture toughness data for chevron notched specimens were smaller than those for the straight cracked samples. For each set of experimental fracture toughness results, probability of fracture was investigated using two- and three-parameter Weibull statistical distributions. Comparison of the Weibull fitted curves for chevron notched and straight cracked samples with the same geometries demonstrated that the discrepancy between the corresponding curves can be described with a good accuracy by a simple shift factor. In addition, using the extended maximum tangential strain criterion which takes into account the influence of both KI and T-stress terms, the statistical fracture toughness data of chevron notched specimens were predicted in terms of the Weibull distribution parameters of the straight cracked specimens.

The statistical distributions of observed flood peaks often show heavy-tailed behaviour, meaning that extreme floods are more likely to occur than for distributions with an exponentially receding tail. Falsely assuming light-tailed behaviour can lead to an underestimation of extreme floods. Robust estimation of the tail is often hindered due to the limited length of time series. Therefore, a better understanding of the processes controlling the tail behaviour is required. Here, we analyse how the spatial variability of rainfall and runoff generation affects the flood peak tail behaviour in catchments of various sizes. This is done using a model chain consisting of a stochastic weather generator, a conceptual rainfall-runoff model, and a river routing routine. For a large synthetic catchment, long time series of daily rainfall with varying tail behaviours and varying degrees of spatial variability are generated and used as input for the rainfall-runoff model. In this model, the spatial variability and mean depth of a sub-surface storage capacity are varied, affecting how locally or widely saturation excess runoff is triggered. Tail behaviour is characterized by the shape parameter of the generalized extreme value (GEV) distribution. Our analysis shows that smaller catchments tend to have heavier tails than larger catchments. For large catchments especially, the GEV shape parameter of flood peak distributions was found to decrease with increasing spatial rainfall variability. This is most likely linked to attenuating effects in large catchments. No clear effect of the spatial variability of the runoff generation on the tail behaviour was found.

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.

Recurrence statistics of large earthquakes has a long-term economic and societal importance. This study investigates the temporal distribution of large (M ≥ 6) earthquakes in the Nepal Himalaya. We compile earthquake data of more than 200 years (1800–2022) and calculate interevent times of successive main shocks. We then derive recurrence-time statistics of large earthquakes using a set of twelve reference statistical distributions. These distributions include the time-independent exponential and time-dependent gamma, lognormal, Weibull, Levy, Maxwell, Pareto, Rayleigh, inverse Gaussian, inverse Weibull, exponentiated exponential and exponentiated Rayleigh. Based on a sample of 38 interoccurrence times, we estimate model parameters via the maximum likelihood estimation and provide their respective confidence bounds through Fisher information and Cramer–Rao bound. Using three model selection approaches, namely the Akaike information criterion (AIC), Kolmogorov–Smirnov goodness-of-fit test and the Chi-square test, we rank the performance of the applied distributions. Our analysis reveals that (i) the best fit comes from the exponentiated Rayleigh (rank 1), exponentiated exponential (rank 2), Weibull (rank 3), exponential (rank 4) and the gamma distribution (rank 5), (ii) an intermediate fit comes from the lognormal (rank 6) and the inverse Weibull distribution (rank 7), whereas (iii) the distributions, namely Maxwell (rank 8), Rayleigh (rank 9), Pareto (rank 10), Levy (rank 11) and inverse Gaussian (rank 12), show poor fit to the observed interevent times. Using the best performed exponentiated Rayleigh model, we observe that the estimated cumulative and conditional occurrence of a M ≥ 6 event in the Nepal Himalaya reach 0.90–0.95 by 2028–2031 and 2034–2037, respectively. We finally present a number of conditional probability curves (hazard function curves) to examine future earthquake hazard in the study region. Overall, the findings provide an important basis for a variety of practical applications, including infrastructure planning, disaster insurance and probabilistic seismic hazard analysis in the Nepal Himalaya.

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.