Pearson System Research Articles

Probabilistic fragility assessment of slopes considering uncertainty associated with temporal patterns of rainfall intensity

The temporal patterns of rainfall intensity, characterized by the variability of rainfall intensity over time during a single rainfall event, play a crucial role in landslide triggering and evolution. However, the influence of temporal patterns of rainfall intensity on the fragility and reliability assessment of slope instability has received limited attention. This study presents a novel methodology for probabilistic slope fragility assessment accounting for uncertainty associated with the temporal patterns of rainfall intensity. Hourly-scale random time series for the rainfall event are generated, considering uncertainties in temporal patterns of rainfall intensity, using the statistical concepts of skewness and kurtosis within the Pearson system of distributions. A time-dependent seepage-stability mechanical model coupling the finite element method and limit equilibrium method for the slope under hourly-scale rainfall time series is established. Slope fragility assessment under rainfall is conducted, accounting for uncertainties in temporal patterns of rainfall intensity based on Monte Carlo simulation. An illustrative example is provided, focusing on a hypothetical soil slope in the southeastern coastal area of China, susceptible to rainfall-induced landslides. The impact of skewness and kurtosis of rainfall time series on the slope failure probability and failure time are examined.

Effects of surface morphology on the performance of water-lubricated thrust bearings

Purpose As the crucial support component of the propeller power system, the reliability of the operation of submersible pumps is influenced by the lubrication performance of water-lubricated thrust bearings. When the water-lubricated thrust bearings are under start-stop or heavy load conditions, the effect of surface morphology is crucial as the mixed lubrication regime is encountered. This paper aims to develop one mixed lubrication model for the water-lubricated thrust bearings to predict the effects of surface skewness, kurtosis and roughness orientation on the loading carrying capacity and tribological behavior. Design/methodology/approach This paper developed one improved mixed lubrication model specifically for the water-lubricated thrust bearing system. In this model, the hydrodynamic model was improved by using the height of the rough surface and its probability density function, combined with the average flow model. The asperity contact model was improved by using the equation for the Pearson system of frequency curves to characterize the non-Gaussian aspect of surface roughness distribution. Findings According to the results, negative skewness, large kurtosis and lateral surface pattern can improve the tribological performance of water-lubricated thrust bearings. Optimizing the surface morphology is a reasonable design method that can improve the performance of water-lubricated thrust bearings. Originality/value In this paper, one mixed lubrication model specifically for the water-lubricated thrust bearing with the effect of surface roughness into consideration was developed. Based on the developed model, the effect of surface morphology on tribological behavior can be evaluated. Peer review The peer review history for this article is available at: https://publons.com/publon/10.1108/ILT-08-2023-0247/

Sparse estimation within Pearson's system, with an application to financial market risk

AbstractPearson's system is a rich class of models that includes many classical univariate distributions. It comprises all continuous densities whose logarithmic derivative can be expressed as a ratio of quadratic polynomials governed by a vector of coefficients. The estimation of a Pearson density is challenging, as small variations in can induce wild changes in the shape of the corresponding density . The authors show how to estimate and effectively through a penalized likelihood procedure involving differential regularization. The approach combines a penalized regression method and a profiled estimation technique. Simulations and an illustration with S&P 500 data suggest that the proposed method can improve market risk assessment substantially through value‐at‐risk and expected shortfall estimates that outperform those currently used by financial institutions and regulators.

ИДЕНТИФИКАЦИЯ ТИПА РАСПРЕДЕЛЕНИЯ НА ПРИМЕРЕ НЕКОТОРЫХ ПОКАЗАТЕЛЕЙ ЦЕННОСТНОГО МИРА РОССИЙСКОЙ МОЛОДЕЖИ

The purpose of the study is to determine the possibility of identifying the type of distribution for the information of questionnaire surveys. The methodological complexity of such a task lies in the fact that the data are essentially unique experiments, for the processing of which it is impossible to use the methods of mathematical statistics. Identification of the distribution type is carried out on the basis of Bayesian intelligent technologies using the Pearson system, according to the methodology developed by one of the authors in the 70s of the last century. The technique has been successfully applied to identification of law distribution in the situations with small samples and under the conditions of considerable uncertainty of information. But for the task of questionnaire data processing this approach is used for the first time. This makes it possible to generate samples of any size according to the obtained distribution law and use them for processing by powerful methods of mathematical statistics. The article provides an example of its successful application to determine the laws of the distribution of indicators of the model of the value world of Russian youth, according to information obtained on the basis of the results of a survey of young people aged 17 to 30 years in 10 subjects of the Russian Federation.

К ВОПРОСУ ОБ ОПРЕДЕЛЕНИИ ПЛОТНОСТИ ВЕРОЯТНОСТИ ТИПОВЫХ РАСПРЕДЕЛЕНИЙ ПО ЭКСПЕРИМЕНТАЛЬНЫМ ДАННЫМ

The article discusses the issues of determining errors in determining the type of distributions. The type of distributions is expressed in the form of probability densities of unimodal distribution laws of the K. Pearson system. Analytical dependences are obtained to determine the necessary volumes of sample data to ensure the required accuracy of identification and approximation of these distribution laws. Specific analytical dependences and graphs are given for determining such distribution laws as simple laws (normal, uniform, triangular) probability distributions, oneparameter laws (gamma distributions, lognormal distribution, Weibull distribution), twoparameter beta distribution.

A new omnibus test of fit based on a characterization of the uniform distribution

ABSTRACT In this paper, we revisit the classical goodness-of-fit problems for univariate distributions; we propose a new testing procedure based on a characterization of the uniform distribution. Asymptotic theory for the simple hypothesis case is provided in a Hilbert-Space setting, including the asymptotic null distribution as well as values for the first four cumulants of this distribution, which are used to fit a Pearson system of distributions as an approximation to the limit distribution. Numerical results indicate that the null distribution of the test converges quickly to its asymptotic distribution, making the critical values obtained using the Pearson system particularly useful. Consistency of the test is shown against any fixed alternative distribution and we derive the limiting behaviour under fixed alternatives with an application to power approximation. We demonstrate the applicability of the newly proposed test when testing composite hypotheses. A Monte Carlo power study compares the finite sample power performance of the newly proposed test to existing omnibus tests in both the simple and composite hypothesis settings. This power study includes results related to testing for the uniform, normal, Pareto and gamma distributions. The empirical results obtained indicate that the test is competitive. An application of the newly proposed test in financial modelling is also included.

Direct tolerance analysis of mechanical assemblies with normal and non-normal tolerances for predicting product quality

ABSTRACT Tolerance analysis, as an effective tool for predicting the effects of geometrical and dimensional deviations on the key characteristics, plays an essential role in increasing the functionality and quality of mechanical products. The conventional methods in the literature have been developed based on this main assumption that the assembly function is available in an explicit form. However, in most industrial applications, deriving an assembly function in an explicit form may be difficult if not impossible. On the other hand, one of the major weaknesses of the conventional tolerance analysis methods in the literature is that all effective dimensions should be varied under the normality assumption. To overcome these weaknesses, in this paper, a new tolerance analysis approach is developed based on the univariate DRM and Pearson system concepts. The proposed method can analyze directly without the need to define any assembly function and also the rejected product rate can be easily predicted using evaluations of the assembly dimension at the limited number of special points. Finally, for verification of the proposed method, some illustrative case studies are considered and results are compared to obtained results of the Monte-Carlo Simulations and the improved Hassofer-Lind and Rack-Fizzler reliability index methods.

Revisiting Empirical Bayes Methods and Applications to Special Types of Data

Empirical Bayes methods have been around for a long time and have a wide range of applications. These methods provide a way in which historical data can be aggregated to provide estimates of the posterior mean. This thesis revisits some of the empirical Bayesian methods and develops new applications. We first look at a linear empirical Bayes estimator and apply it on ranking and symbolic data. Next, we consider Tweedie's formula and show how it can be applied to analyze a microarray dataset. The application of the formula is simplified with the Pearson system of distributions. Saddlepoint approximations enable us to generalize several results in this direction. The results show that the proposed methods perform well in applications to real data sets.

Empirical Bayes on a shoestring and other applications

In the article, we exploit the Pearson system of distributions to obtain expressions for the posterior mean and posterior variance in Tweedie’s formula. This has interesting consequences in the study of micro-array data where the interest is uncovering unusual genes. Another application deals with the development of new non-parametric tests for the two sample problems of location and of scale. These tests are based on an approximation of the locally optimal score test statistics.

On combining the zero bias transform and the empirical characteristic function to test normality

We propose a new powerful family of tests of univariate normality. These tests are based on an initial value problem in the space of characteristic functions originating from the fixed point property of the normal distribution in the zero bias transform. Limit distributions of the test statistics are provided under the null hypothesis, as well as under contiguous and fixed alternatives. Using the covariance structure of the limiting Gaussian process from the null distribution, we derive explicit formulas for the first four cumulants of the limiting random element and apply the results by fitting a distribution from the Pearson system. A comparative Monte Carlo power study shows that the new tests are serious competitors to the strongest well established tests.

A novel hybrid cubature formula with Pearson system for efficient moment-based uncertainty propagation analysis

In this paper, a novel hybrid cubature formula is proposed for moment-based uncertainty propagation analysis. First, the contribution-degree analysis is performed to classify the input random vector of the response function into two separate parts, i.e. the more important one and less important one. In this regard, the statistical moment of the response function, which is a multi-dimensional Gaussian-weighted integral, can be decomposed into one lower-dimensional Gaussian-weighted integral and several one-dimensional Gaussian-weighted integrals, accordingly. Then, the hybrid cubature formula can be established for the first-four statistical moments assessment of the response function such that a mixed-degree cubature formula is employed to evaluate the lower-dimensional integral and the five-point Gauss-Hermite quadrature is adopted for obtaining the one-dimensional integrals. By doing so, the trade-off of accuracy and efficiency for statistical moments assessment can be ensured. Finally, the Pearson system is employed to reconstruct the probability density function of the response function. Five numerical examples are investigated to demonstrate the performance of the proposed method for uncertainty propagation analysis, where pertinent Monte Carlo simulations are also conducted for comparisons. It is found that the proposed method can achieve a good accuracy for evaluating the first-four central moments of the response function as well as its entire distribution of the probability density function with high efficiency.

ПРИМЕНЕНИЕ РЕГУЛЯРИЗИРУЮЩЕГО БАЙЕСОВСКОГО ПОДХОДА ДЛЯ ОПРЕДЕЛЕНИЯ ЗАКОНОВ РАСПРЕДЕЛЕНИЯ РЕЙТИНГОВЫХ ОЦЕНОК БАНКОВ

Determining the rating ratings of banks is important both for the banks themselves and for assessing the state of the entire financial sector of the state. Their compliance with real situations depends on the calculation models used by rating agencies. One of the main stages of calculation in the development of the model is the stage of determining the ranges of estimates. In many methods of rating banks, the ranges are determined by experts and are subjective in nature. In this paper, we propose an algorithm for determining the ranges of rating ratings using the approximation of the distribution laws of the estimated indicators of the model by the Pearson system curves. This approach can significantly improve the accuracy and objectivity of rating assessments

Numerical study of normal contact stiffness: Non-Gaussian roughness and elastic–plastic behavior

In this paper, we have studied the effect of the degree of asymmetry and peakedness on the performances of a rough contact. A numerical model is proposed to evaluate the real area, the number of contacting asperities and the normal stiffness. The contact between two rough surfaces was modeled as a contact between a rigid flat and an elastic–plastic rough surface taking into account the work hardening and interaction among asperities. The Pearson system of frequency curves was used to generate non-Gaussian random surfaces. Results show that a negative skewness and/or a low kurtosis improve the contact performances by increasing the normal stiffness. This increase is governed by much more asperities into contact and more elastic behavior.

Computing and graphing probability values of pearson distributions: a SAS/IML macro

BackgroundAny empirical data can be approximated to one of Pearson distributions using the first four moments of the data (Elderton WP, Johnson NL. Systems of Frequency Curves. 1969; Pearson K. Philos Trans R Soc Lond Ser A. 186:343–414 1895; Solomon H, Stephens MA. J Am Stat Assoc. 73(361):153–60 1978). Thus, Pearson distributions made statistical analysis possible for data with unknown distributions. There are both extant, old-fashioned in-print tables (Pearson ES, Hartley HO. Biometrika Tables for Statisticians, vol. II. 1972) and contemporary computer programs (Amos DE, Daniel SL. Tables of percentage points of standardized pearson distributions. 1971; Bouver H, Bargmann RE. Tables of the standardized percentage points of the pearson system of curves in terms of β1 and β2. 1974; Bowman KO, Shenton LR. Biometrika. 66(1):147–51 1979; Davis CS, Stephens MA. Appl Stat. 32(3):322–7 1983; Pan W. J Stat Softw. 31(Code Snippet 2):1–6 2009) available for obtaining percentage points of Pearson distributions corresponding to certain pre-specified percentages (or probability values; e.g., 1.0%, 2.5%, 5.0%, etc.), but they are little useful in statistical analysis because we have to rely on unwieldy second difference interpolation to calculate a probability value of a Pearson distribution corresponding to a given percentage point, such as an observed test statistic in hypothesis testing.ResultsThe present study develops a SAS/IML macro program to identify the appropriate type of Pearson distribution based on either input of dataset or the values of four moments and then compute and graph probability values of Pearson distributions for any given percentage points.ConclusionsThe SAS macro program returns accurate approximations to Pearson distributions and can efficiently facilitate researchers to conduct statistical analysis on data with unknown distributions.

Modeling for Microslip Behavior of Lap Joints Based on Non-Gaussian Rough Surfaces

AbstractA general contact model for a lap joint interface based on non-Gaussian surfaces was proposed. The effect of surface topography parameters on microslip behavior in a lap joint interface was studied. Pearson system was applied to produce non-Gaussian surfaces. Combining the topographical-dependent Zhao–Maietta–Chang (ZMC) model with the physical-related Iwan model, the nonlinear constitutive relationship of a lap interface was constructed by using Masing hypothesis. Meanwhile, the probability density function of asperity heights of an infinitely smooth surface was mathematically proved to be a delta function, verifying that the calculated value of friction in the model conforms to the physical law. Gauss-Legendre quadrature was conducted to calculate contact relations of different Pearson distribution surfaces. Furthermore, numerical results of microslip loops under oscillating tangential forces were compared with the published experiments, indicating the present model considering non-Gaussian surfaces could agree well with the experiments.

Reliability based design optimization using response surface augmented moment method

A new approach toward reliability based design optimization (RBDO) is proposed based on the response surface augmented moment method (RSMM). In RSMM, the reliability analysis procedure based on design of experiments (DOE) is combined with the response surface method (RSM) for numerical efficiency. It utilizes the Pearson system with four statistical moments to calculate the failure probability, and the progressive update of the response surface facilitates the calculation of these four statistical moments. In this study, a semi-analytic design sensitivity analysis is performed in connection with RSMM for an efficient implementation of RSMM in RBDO. The sensitivity of failure probability with respect to the design variables is calculated by direct differentiation and finite difference method with the Pearson system. It is integrated into a mathematical programming for RBDO and applied to several test examples. It was demonstrated that the proposed method of RBDO based on RSMM is efficient and robust.

Selecting the Probability Distribution of Cone Tip Resistance Using Moment Ratio Diagram for Soil in Nasiriyah

Selecting suitable probability distributions (PDs) to describe cone tip resistance measurements (qc) obtained by a cone penetration test (CPT) is considered a crucial requirement to get a good solution for geotechnical problems solved by simulating the engineering properties of soil as a random field or for use in reliability-based design. This paper presents a statistical analysis of seven PDs proposed to model qc obtained through performing CPT for soil in Nasiriyah during the construction of a new refinery petrol station. Preliminary testing for suitability of the suggested distributions has used the method of moment ratio diagram (MRD) based on the Pearson system. It was found that the soil stratification has a large effect on the distance between every two points on MRD. The type of probability distribution was also affected, and changed, by increasing the number of data points for qc included in the analysis. Logistic and Weibull distributions are considered the best PDs that represent the qc of the first layer having thickness 12 m of clay soil, followed by the other distributions, while the logistic and normal distributions were considered the best PDs among the seven suggested distributions for the second layer of 8 m silty sand and clayey sand. All the suggested distribution can represent the given qc data approximately except the Rayleigh distribution.

Modeling tangential contact based on non-Gaussian rough surfaces

A stick-slip tangential contact model between rough surfaces was proposed in this paper. The Mindlin partial slip solution for elastic contact and the double linear formulation developed by Fujimoto for plastic contact in conjunction with the Coulomb dry friction law were used to derive tangential contact formulas. Pearson system of frequency curves was used to generate non-Gaussian random surfaces. Effects of skewness and kurtosis on normal and tangential contact responses were studied independently. The results showed that negative skewness predicted lower mean separation for a given normal force and greater tangential stiffness, while for positive skewness, there exist different trends from negative skewness. With the increase of kurtosis, the load capacity and tangential stiffness decreased. The practical significance of these findings is that it can help engineers to design proper surface textures based on their requirements. A comparison of initial tangential stiffness between predicted results and published experimental results was made. The results well agreed with the experimental results when the non-Gaussian surface effects were taken into consideration.

Some Ghosts of Statistics Past

Some Ghosts of Statistics Past

The effects of sector reforms on the productivity of Greek banks: a step-by-step analysis of the pre-Euro era

The paper analyses the effects on the productivity of Greek commercial banks of sector regulatory reforms in the pre-Euro era, using the Global Malmquist Index. In a bootstrap Data Envelopment Analysis framework, we propose an alternative to smoothing that utilises the Pearson system random number generator, offering greater flexibility in the choice of the fitting distribution. In the context of a step-by-step approach, we demonstrate the contribution of deregulatory commercial freedoms to greater productivity and the negative effect of prudential controls. Our findings offer insights into the current state of the Greek banking sector, suggesting that the imposition of additional prudential controls may have a detrimental impact on the productivity of Greek banks, given the adverse business conditions.