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We consider random orthonormal polynomials \begin{equation*} F_{n}(x)=\sum _{i=0}^{n}\xi _{i}p_{i}(x), \end{equation*} where \xi _{0}, ..., \xi _{n} are independent random variables with zero mean, unit variance and uniformly bounded (2+\varepsilon) moments, and (p_n)_{n=0}^{\infty } is the system of orthonormal polynomials with respect to a fixed compactly supported measure on the real line. Under mild technical assumptions satisfied by many classes of classical polynomial systems, we establish universality for the leading asymptotics of the average number of real roots of F_n, both globally and locally. Prior to this paper, these results were known only for random orthonormal polynomials with Gaussian coefficients (see D. D. Lubinsky, I. E. Pritsker, and X. Xie [Proc. Amer. Math. Soc. 144 (2016), pp. 1631–1642]) using the Kac-Rice formula, a method that does not extend to the generality of our paper. [ABSTRACT FROM AUTHOR]

In the current paper, we review existing tools for solving variable selection problems in psychology. Modern regularization methods such as lasso regression have recently been introduced in the field and are incorporated into popular methodologies, such as network analysis. However, several recognized limitations of lasso regularization may limit its suitability for psychological research. In this paper, we compare the properties of lasso approaches used for variable selection to Bayesian variable selection approaches. In particular we highlight advantages of stochastic search variable selection (SSVS), that make it well suited for variable selection applications in psychology. We demonstrate these advantages and contrast SSVS with lasso type penalization in an application to predict depression symptoms in a large sample and an accompanying simulation study. We investigate the effects of sample size, effect size, and patterns of correlation among predictors on rates of correct and false inclusion and bias in the estimates. SSVS as investigated here is reasonably computationally efficient and powerful to detect moderate effects in small sample sizes (or small effects in moderate sample sizes), while protecting against false inclusion and without over-penalizing true effects. We recommend SSVS as a flexible framework that is well-suited for the field, discuss limitations, and suggest directions for future development. [ABSTRACT FROM AUTHOR]

This paper considers a general stochastic resource allocation problem that arises widely in wireless networks, cognitive radio, networks, smart-grid communications, and cross-layer design. The problem formulation involves expectations with respect to a collection of random variables with unknown distributions, representing exogenous quantities such as channel gain, user density, or spectrum occupancy. We consider the constant step-size stochastic dual subgradient descent (SDSD) method that has been widely used for online resource allocation in networks. The problem is solved in dual domain, which results in a primal resource allocation subproblem at each time instant. The goal here is to characterize the non-asymptotic behavior of such stochastic resource allocations in an almost sure sense. It is well known that with a step size of \epsilon , SDSD converges to an \mathcal {O}(\epsilon) -sized neighborhood of the optimum. In practice, however, there exists a trade-off between the rate of convergence and the choice of \epsilon $ . This paper establishes a convergence rate result for the SDSD algorithm that precisely characterizes this trade-off. Toward this end, a novel stochastic bound on the gap between the objective function and the optimum is developed. The asymptotic behavior of the stochastic term is characterized in an almost sure sense, thereby generalizing the existing results for the stochastic subgradient methods. For the stochastic resource allocation problem at hand, the result explicates the rate with which the allocated resources become near-optimal. As an application, the power and user-allocation problem in device-to-device networks is formulated and solved using the SDSD algorithm. Further intuition on the rate results is obtained from the verification of the regularity conditions and accompanying simulation results. [ABSTRACT FROM PUBLISHER]

In this paper, we study bounds for the total variation distance between distributions of second order polynomials in normal random variables provided that they essentially depend on at least three variables. Moreover, we partially extend some recent bounds for the Kolmogorov distance between the distributions of norms of Gaussian random vectors to the case of the total variation distance. [ABSTRACT FROM AUTHOR]

With the in-depth development of social reforms, the scientificization of enterprise online examinations has become more and more urgent and important. The key to realizing scientific examinations is the automation and rationalization of propositions. Therefore, the construction and realization of the test question bank is also more important. In the realization of the entire test question database, how to select satisfactory test questions randomly from a large number of test questions through the selection of test questions so that the average difficulty, discriminability, and reliability of the test are satisfactory? These requirements are also more important. Among them, random selection of questions is an important difficulty in the realization of the test question bank. In order to solve the difficulties of random selection of these test questions, the author combines the experience of constructing the test question bank and uses the discrete binomial distribution to draw conclusions. Random variables established the first mathematical model for topic selection. By determining the form of the test questions and the distribution of the difficulty of the test questions and then making it use a random function to select questions, this will achieve better results. [ABSTRACT FROM AUTHOR]

In this paper, under some suitable assumptions, using the Taylor expansion, Borel–Cantelli lemma and the almost sure central limit theorem for independent random variables, the almost sure central limit theorem for error variance estimator in the pth-order nonlinear autoregressive processes with independent and identical distributed errors was established. Four examples, first-order autoregressive processes, self-exciting threshold autoregressive processes, threshold-exponential AR progresses and multilayer perceptrons progress, are given to verify the results. [ABSTRACT FROM AUTHOR]

In this paper, we assume that cause–effect relationships between random variables can be represented by a Gaussian linear structural equation model and the corresponding directed acyclic graph. Then, we consider a situation where a set of random variables that satisfies the front-door criterion is observed to estimate a total effect. In this situation, when the ordinary least squares method is utilized to estimate the total effect, we formulate the unbiased estimator of the causal effect on the variance of the outcome variable. In addition, we provide the exact variance formula of the proposed unbiased estimator. [ABSTRACT FROM AUTHOR]

In this paper, we obtain an approximation for the distribution function of sum of two independent random variables using quantile based representation. The error of approximation is shown to be negligible under some mild conditions. We then use the approximation to obtain a non-parametric estimator for the distribution function of sum of two independent random variables. The exact distribution of the proposed estimator is derived. The estimator is shown to be strongly consistent and asymptotically normally distributed. Extensive Monte Carlo simulation studies are carried out to evaluate the bias and mean squared error of the estimator and also to assess the approximation error. We also compare the performance of the proposed estimator with other estimators available in the literature. Finally, we illustrate the use of the proposed estimator for estimating the reliability function of a standby redundant system. [ABSTRACT FROM AUTHOR]

Computational statistics is a critical skill for professionals in fields such as data science, statistics, and related disciplines. One essential aspect of computational statistics is the ability to simulate random variables from specified probability distributions. Commonly employed techniques for sampling random variables include the inverse transform method, acceptance–rejection method, and Box–Muller transformation, all of which rely on sampling from the uniform (0 , 1) distribution. A significant concept in statistics is the finite mixture model, characterized by a convex combination of multiple probability density functions. In this paper, we introduce a modified version of the composition method, a standard approach for sampling finite mixture models. Our modification offers the advantage of relying on sampling from the uniform (0 , 1) distribution, aligning with prevalent methods in computational statistics. This alignment simplifies teaching computational statistics courses, as well as having other benefits. We offer several examples to illustrate the approach. [ABSTRACT FROM AUTHOR]

The anticipative information refers to some information about future events that may be disclosed in advance. This information may regard, for example, financial assets and their future trends. In our paper, we assume the existence of some anticipative information in a market whose risky asset dynamics evolve according to a Brownian motion and a Poisson process. Using Malliavin calculus and filtration enlargement techniques, we derive the information drift of the mentioned processes and, both in the pure jump case and in the mixed one, we compute the additional expected logarithmic utility. Many examples are shown, where the anticipative information is related to some conditions that the constituent processes or their running maximum may verify, in particular, we show new examples considering Bernoulli random variables. [ABSTRACT FROM AUTHOR]

Quantum-behaved Particle Swarm Optimization (QPSO) is a meta-heuristic optimization algorithm, which is widely used in many research fields and practical problems due to its flexibility and low computational cost. However, the existing QPSO algorithms and their variants still have problems such as insufficient search capabilities, lack of adaptivity, and prone to stagnation. This paper proposes a novel ensemble algorithm, ACQPSOW-HGS, based on Quantum-behaved Particle Swarm Optimization (QPSO) and Hunger Games Search (HGS). By combining three improvements and introducing three hybrid strategies, our algorithm has made a comprehensive development, effectively improving the stability and solution accuracy in a large number of test functions and the parameter identification application, which is superior compared with many existing algorithms. First, we design the Weibull distribution random number generation operator, the distance-guided adaptive control technique, and the chaotic update mechanism to deal with the weak randomness, insufficient adaptability, and susceptibility to stagnation of the original QPSO algorithm, respectively. Integrating the above improvements, ACQPSOW is proposed as an improved variant of QPSO. Second, the proposed ensemble algorithm ACQPSOW-HGS is built on ACQPSOW and HGS and combined with specific hybrid strategies to add population diversity and improve search efficiency, including the Selection-Crossover-Mutation mechanism, the elite local search mechanism, and the information exchange mechanism. Finally, the experiments, on 23 benchmark functions and the IEEE CEC 2017 test suite, demonstrate that ACQPSOW-HGS outperforms comparison algorithms in terms of convergence speed and solution accuracy through non-parametric statistical tests. Moreover, ACQPSOW-HGS was applied to the fractional-order hyper-chaotic financial system for parameter identification to illustrate the applicability and robustness in solving real-world problems. [ABSTRACT FROM AUTHOR]

In time-variant reliability problems, there are a lot of uncertain variables from different sources. Therefore, it is important to consider these uncertainties in engineering. In addition, time-variant reliability problems typically involve a complex multilevel nested optimization problem, which can result in an enormous amount of computation. To this end, this paper studies the time-variant reliability evaluation of structures with stochastic and bounded uncertainties using a mixed probability and convex set model. In this method, the stochastic process of a limit-state function with mixed uncertain parameters is first discretized and then converted into a time-independent reliability problem. Further, to solve the double nested optimization problem in hybrid reliability calculation, an efficient iterative scheme is designed in standard uncertainty space to determine the most probable point (MPP). The limit state function is linearized at these points, and an innovative random variable is defined to solve the equivalent static reliability analysis model. The effectiveness of the proposed method is verified by two benchmark numerical examples and a practical engineering problem. [ABSTRACT FROM AUTHOR]

In general, the extended Kalman filter (EKF) has a wide range of applications, aiming to minimize symmetric loss function (mean square error) and improve the accuracy and efficiency of state estimation. As the nonlinear model complexity increases, rounding errors gradually amplify, leading to performance degradation. After multiple iterations, divergence may occur. The traditional extended Kalman filter cannot accurately estimate the nonlinear model, and these errors still have an impact on the accuracy. To improve the filtering performance of the extended Kalman filter (EKF), this paper proposes a new extended Kalman filter (REKF) method that utilizes the statistical properties of the rounding error to enhance the estimation accuracy. After establishing the state model and measurement model, the residual term is used to replace the higher-order term in the Taylor expansion, and the least squares method is applied to identify the residual term step by step. Then, the iterative process of updating the extended Kalman filter is carried out. Within the Kalman filter framework, a higher-order rounding error-based extended Kalman filter (REKF) is designed for the joint estimation of rounding error and random variables, and the solution method for the rounding error is considered for the multilevel approximation of the original function. Through numerical simulations on a general nonlinear model, the higher-order rounding error-based extended Kalman filter (REKF) achieves better estimation results than the extended Kalman filter (EKF) and improves the filtering accuracy by utilizing the higher-order rounding error information, which also proves the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we have considered the classical and Bayesian inference of the unknown parameters of the lifetime distribution based on the observations obtained from a simple step‐stress life‐testing (SSLT) experiment and when more than one cause of failures are observed. We have used the Tampered Random Variable (TRV) approach. The main advantage of the TRV approach is that it can be easily extended to a multiple step‐stress model as well as for different lifetime distributions. In this paper, it is assumed that the lifetime of the experimental units at each stress level follows Weibull distribution with the same shape parameter and different scale parameters. Further, we have introduced different tempering co‐efficient for different causes of failures. The maximum likelihood estimators and the associated asymptotic confidence intervals are obtained based on Type‐II censored observations. Further, we have considered the Bayesian inference of the unknown model parameters based on a fairly general class prior distributions. An extensive simulation study is performed to examine the performances of the proposed method, and the analysis of a real data set has been provided to show how the method can be used in practice. We have compared the TRV model with some of the other existing models, and the TRV model provides a better fit in terms of information theoretic criteria. We have also provided some optimality criteria, to determine the optimal stress change time and some sensitivity analyses have been performed. Most of the methods can be extended for other lifetime distributions also. [ABSTRACT FROM AUTHOR]

In the conditions of technological progress and rapid development of information technologies, it is very challenging to prepare specialists in the field of information and communication technologies who would fully possess the knowledge and skills to work with the diverse range of instrumental and software tools and equipment used in the field. This is due to the continuous improvement and updating of programming languages, frameworks, software environments, shells, and hardware. One of the ways to address this issue is through computer-based educational programs, allowing the individualization of the learning process while preserving it’s quality. Currently, there is a large number of computer-based educational programs used in the study of various disciplines. This paper presents the design of a computer-based educational program on the subject "Fundamentals of Computer Modeling," which enables the modeling of random patterns of various natures, ranging from random events to random streams. The educational program consists of three modules, which include a theoretical part on methods of modeling random patterns, testing after each topic, and a calculator for all modeling methods. In each modeling method, you can input any initial data, such as the number of trials, probabilities of events, functions, and parameters of distribution laws. The results of modeling random patterns can be saved as a file. The program interface is available in three languages: Russian, English, and Kazakh. The currently developing educational program aims to improve the learning process in the "Fundamentals of Computer Modeling" discipline and allows learners to study it independently without any interventions from a professor. [ABSTRACT FROM AUTHOR]

Matroids and semigraphoids are discrete structures abstracting and generalizing linear independence among vectors and conditional independence among random variables, respectively. Despite the different nature of conditional independence from linear independence, deep connections between these two areas are found and are still undergoing active research. In this paper, we give a characterization of the embedding of matroids into conditional independence structures and its oriented counterpart, which leads to new axiom systems of matroids and oriented matroids. [ABSTRACT FROM AUTHOR]

In this paper, the displacement variation in sandwich beams is determined by employing a semi-analytical approach. The classical displacement is calculated by integration using Mohr's equation, although the integration is complicated due to the inclusion of random fields in the inertial moment term. Using the trapezoidal rule to compute these integrals, the random fields are discretized into random variables at the nodal point of the beam segments. Thus, the expected displacement, standard deviation, and coefficient of variation can be computed. To validate the results, the random fields are simulated using a previously described spectral method. The results of numerical examples were compared with the semi-analytical method and the Monte Carlo simulation demonstrating the high accuracy of the proposed method. The results also illustrate the influence of the parameters of the random fields of elastic modulus on the variability of displacement. [ABSTRACT FROM AUTHOR]

Demand-side flexible load resources, such as Electric Vehicles (EVs) and Air Conditioners (ACs), offer significant potential for enhancing flexibility in the power system, thereby promoting the full integration of renewable energy. To this end, this paper proposes an optimal allocation method for demand-side flexible resources to enhance renewable energy consumption. Firstly, the adjustable flexibility of these resources is modeled based on the generalized energy storage model. Secondly, we generate random scenarios for wind, solar, and load, considering variable correlations based on non-parametric probability predictions of random variables combined with Copula function sampling. Next, we establish the optimal allocation model for demand-side flexible resources, considering the simulated operation of these random scenarios. Finally, we optimize the demand-side resource transformation plan year by year based on the growth trend forecast results of renewable energy installed capacity in Jiangsu Province from 2025 to 2031. [ABSTRACT FROM AUTHOR]

In this paper, we consider a tandem dual queuing system consisting of multi-server stages. Stage 1 is characterized by an infinite buffer, one-by-one service of customers, and an exponential distribution of service times. Stage 2 is characterized by a finite buffer and a phase-type distribution of service times. Service at Stage 2 is provided to groups of customers. The service time of a group depends on the size of the group. The size is restricted by two thresholds. The waiting time of a customer at each stage is limited by a random variable with an exponential distribution, with the parameter depending on the stage. After service at Stage 1, a customer can depart from the system or try to enter Stage 2. If the buffer at this stage is full, the customer is either lost or returns for service at Stage 1. Customer arrivals are described by the versatile Markov arrival process. The system is studied via consideration of a multi-dimensional continuous-time Markov chain. Numerical examples, which highlight the influence of the thresholds on the system performance measures, are presented. The possibility of solving optimization problems is illustrated. [ABSTRACT FROM AUTHOR]

The main focus of this paper is the derivation of the structural properties of the test channels of Wyner's operational information rate distortion function (RDF), R ¯ (Δ X) , for arbitrary abstract sources and, subsequently, the derivation of additional properties for a tuple of multivariate correlated, jointly independent, and identically distributed Gaussian random variables, { X t , Y t } t = 1 ∞ , X t : Ω → R n x , Y t : Ω → R n y , with average mean-square error at the decoder and the side information, { Y t } t = 1 ∞ , available only at the decoder. For the tuple of multivariate correlated Gaussian sources, we construct optimal test channel realizations which achieve the informational RDF, R ¯ (Δ X) = ▵ inf M (Δ X) I (X ; Z | Y) , where M (Δ X) is the set of auxiliary RVs Z such that P Z | X , Y = P Z | X , X ^ = f (Y , Z) , and E { | | X − X ^ | | 2 } ≤ Δ X . We show the following fundamental structural properties: (1) Optimal test channel realizations that achieve the RDF and satisfy conditional independence, P X | X ^ , Y , Z = P X | X ^ , Y = P X | X ^ , E X | X ^ , Y , Z = E X | X ^ = X ^. (2) Similarly, for the conditional RDF, R X | Y (Δ X) , when the side information is available to both the encoder and the decoder, we show the equality R ¯ (Δ X) = R X | Y (Δ X) . (3) We derive the water-filling solution for R X | Y (Δ X) . [ABSTRACT FROM AUTHOR]

In reliability theory, the kth order statistic represents the lifetime of a (n - k + 1) -out-of-n system. In particular, the smallest (largest) order statistics denotes the lifetime of a series (parallel) system that consist of n components; and the second smallest (second largest) order statistics is the lifetime of a (n - 1) -out-of-n (2-out-of-n) system. In this paper, sufficient conditions are provided to compare the smallest and the second smallest (largest and second largest) order statistics of dependent and heterogeneous random variables having the additive hazard model with the Archimedean copula in the sense of usual stochastic order and hazard rate order. Further, we compare the smallest order statistics of two sets of independent and heterogeneous random variables having the additive hazard model in the sense of dispersive order. Finally, our theoretical findings are evaluated by some numerical examples and counterexamples. [ABSTRACT FROM AUTHOR]

Extreme waves pose one of the major threats to marine structures. Furthermore, their non-stationary nature makes proper stochastic analysis of their responses a challenging problem. To address this issue, this paper proposes a method based on a novel linear filter wave model incorporated into the dimension-reduced probability density evolution equation (DR-PDEE). The linear filter system is capable of simulating random background waves conforming with the Joint North Sea Wave Project (JONSWAP) spectrum of any arbitrary sea state by adjusting the parameters of filters directly related to the parameters of the JONSWAP spectrum without reidentification. In particular, by conducting the digital filtering, wave kinematics at different depths below the sea surface can be reproduced conveniently, and therefore only one filter is adequate for the depthwise wave kinematics field. Extreme ocean waves are treated as the superposition of background waves and extreme crests according to the constrained quasi-determinism method, with randomness from both parts. Incorporating the filter into the equation of motion of the offshore structure of interest leads to an augmented high-dimensional stochastic system with multiple random variables. The DR-PDEE then is employed to reduce the dimensions of the equation governing the evolution of probability density of responses of the original complex system to two. Solving the DR-PDEE using the path integral method yields the probability function of the response at each time step. A numerical example involving the response of a National Renewable Energy Laboratory (NREL) 5-MW spar-type floating offshore wind turbine (FOWT) subjected to extreme waves was studied to assess the reliability of the proposed method. The method provides an effective tool for the determination of the stochastic extreme response of offshore structures, and provides a foundation for further dynamic analyses. [ABSTRACT FROM AUTHOR]

In the face of increasingly complex data forms and decision-making problems, the uncertainty of information poses a major challenge to multi-attribute decision-making methods. How to effectively organize information and serve realistic decision-making problems has attracted extensive attention in the academic circles. In view of this, based on the distribution law of random variables, we put forward the basic concept of probability numbers and construct a general framework, including the concepts of type, order, item, isomorphism and isomerism, same domain and same distribution of probability numbers. On this basis, we further define the expectation and variance formula of probability numbers, and its operation rules are defined for the same type of probability numbers. To compare the dominance and inferiority of probability numbers further accurately, we put forward the concepts of dominance degree and comparability degree of probability numbers, so that decision makers can realize the ranking of probability numbers by calculating the comprehensive dominance degree. In view of the related concepts of probability numbers, we summarize the properties and theorems of probability numbers and prove them. In addition, a probability numbers-based multi-attribute decision-making framework model is proposed to solve the multi-attribute decision-making problem. Decision makers can select appropriate sub-models to construct personalized multi-attribute decision-making methods according to actual needs. At the end of the paper, we apply the method to the multi-attribute decision case of campus express stations evaluation and verify the scientificity and rationality of the evaluation method. The concept of probability numbers and its decision model proposed in this paper extend the concept category of numbers, enrich the multi-attribute decision-making method based on probability numbers, and have certain reference significance for further research of uncertain decision theory and method. [ABSTRACT FROM AUTHOR]

In this paper, we study three numerical schemes for the McKean-Vlasov equation {dXt=b(t,Xt,μt)dt+σ(t,Xt,μt)dBt,∀t∈[0,T],μt is the probability distribution of Xt, { d X t = b (t , X t , μ t) d t + σ (t , X t , μ t) d B t , ∀ t ∈ [ 0 , T ] , μ t  is the probability distribution of   X t , $ \{ \begin{array}{l}dX_t=b{(t,X_t,\mu_t)}dt+\sigma{(t,X_t,\mu_t)}dB_t,\\ \forall t\in\lbrack 0,T\rbrack,\mu_t\text{is the probability distribution of }X_t,\end{array} $ where X0 : (Ω, F, ℙ) → (ℝd, B(ℝd)) is a known random variable. Under the assumption on the Lipschitz continuity of the coefficients b and σ, our first result proves the convergence rate of the particle method with respect to the Wasserstein distance, which extends previous work [M. Bossy and D. Talay, Math. Comput. 66 (1997) 157–192.] established in a one-dimensional setting. In the second part, we present and analyse two quantization-based schemes, including the recursive quantization scheme (deterministic scheme) in the Vlasov setting, and the hybrid particle-quantization scheme (random scheme inspired by the K-means clustering). Two simulations are presented at the end of this paper: Burgers equation introduced in [M. Bossy and D. Talay, Math. Comput. 66 (1997) 157–192.] and the network of FitzHugh- Nagumo neurons (see [J. Baladron, D. Fasoli, O. Faugeras and J. Touboul, J. Math. Neurosci. 2 (2012) 1–50.] and [M. Bossy, O. Faugeras and D. Talay, J. Math. Neurosci. 5 (2015) 1–23.]) in dimension 3. [ABSTRACT FROM AUTHOR]

In this paper, the seven equivalent conditions of complete moment convergence and complete integral convergence for m -asymptotic negatively associated ( m -ANA, for short) random variables are established. The results obtained in the paper extend and improve some corresponding ones for negatively associated (NA, for short) random variables and negatively orthant dependent (NOD, for short) random variables. As an application of our main results, we present a result on complete consistency for the weighted estimator in a nonparametric regression model based on m-ANA errors. We perform a numerical simulation to verify the validity of the theoretical results. [ABSTRACT FROM AUTHOR]

Dynamic reliability evaluation of large-scale reinforced concrete (RC) structures is one of the most challenging problems in engineering practices. Although extensive endeavors have been devoted to mechanical analysis of concrete structures in the past decades, it was recognized that the randomness from both structural parameters and excitations have significant effects on the dynamic behaviors of structures with complex nonlinearity, damage, energy dissipation, and plasticity. Thus, great difficulty exists in evaluating the nonlinear stochastic responses and dynamic reliability of the real-world complex structures of large degrees of freedom. In the present paper, a physically driven method for refined probabilistic response and seismic reliability evaluation of real-world RC structures is proposed via synthesis of the refined mechanical analysis and the physically-based uncertainty propagation. In this method, the material parameters can be treated as probabilistically dependent random variables characterized by vine copulas, and the ground motion is modeled by non-stationary Clough–Penzien spectrum. The uncertainty propagation of arbitrary response quantity(-ies) of interest is governed by the dimension-reduced probability density evolution equation (DR-PDEE). The intrinsic drift coefficients in the DR-PDEE are the physically driven force for the uncertainty propagation and can be identified via data from representative dynamic analyses of the structure. The time-variant reliability of the structures can be captured by solving the physically driven DR-PDEE, which cannot be achieved by the general Monte Carlo simulation (MCS) due to the prohabitively large computational cost. Finally, a practical engineering application is shown in this paper for the probabilistic response and seismic reliability evaluation of a 24-story RC shear wall structure with nearly 280,000 degrees of freedom (DOFs). [ABSTRACT FROM AUTHOR]

In this paper, we first study doubly truncated (interval) Tsallis entropy and suggest doubly truncated (interval) cumulative residual Tsallis entropy (ICRT), which is an extension of cumulative residual Tsallis entropy (CRT) and the dynamic CRT defined by the aid of Sati and Gupta and of Kumar, respectively. We investigate some properties and characterization of this measure, such as its relation with doubly truncated Shannon entropy, mean residual (past) life, and hazard rate (or reversed hazard rate). Also, the twin measure, doubly truncated (interval) cumulative past Tsallis entropy, is determined, and some of its properties are studied. Moreover, their monotonicity and related aging classes of distributions are expressed, and the upper (lower) bound for them is acquired. In the end, we propose four nonpara-metric estimators and compare their performance by utilizing simulation data. Also, being based on the best-proposed estimator, a real data set is additionally examined. [ABSTRACT FROM AUTHOR]

In this paper, we prove an extension of the Jajte strong law of large numbers for exchangeable random variables, we make a simulation study for the asymptotic behavior in the sense of convergence almost surly for weighted sums of exchangeable weighted random variables. [ABSTRACT FROM AUTHOR]

Partial differential equations (PDEs) with uncertain or random inputs have been considered in many studies of uncertainty quantification. In forward uncertainty quantification, one is interested in analyzing the stochastic response of the PDE subject to input uncertainty, which usually involves solving high-dimensional integrals of the PDE output over a sequence of stochastic variables. In practical computations, one typically needs to discretize the problem in several ways: approximating an infinite-dimensional input random field with a finite-dimensional random field, spatial discretization of the PDE using, e.g., finite elements, and approximating high-dimensional integrals using cubatures such as quasi--Monte Carlo methods. In this paper, we focus on the error resulting from dimension truncation of an input random field. We show how Taylor series can be used to derive theoretical dimension truncation rates for a wide class of problems and we provide a simple checklist of conditions that a parametric mathematical model needs to satisfy in order for our dimension truncation error bound to hold. Some of the novel features of our approach include that our results are applicable to nonaffine parametric operator equations, dimensionally truncated conforming finite element discretized solutions of parametric PDEs, and even compositions of PDE solutions with smooth nonlinear quantities of interest. As a specific application of our method, we derive an improved dimension truncation error bound for elliptic PDEs with lognormally parameterized diffusion coefficients. Numerical examples support our theoretical findings. [ABSTRACT FROM AUTHOR]

The partial information decomposition (PID) framework is concerned with decomposing the information that a set of (two or more) random variables (the sources) has about another variable (the target) into three types of information: unique, redundant, and synergistic. Classical information theory alone does not provide a unique way to decompose information in this manner and additional assumptions have to be made. One often overlooked way to achieve this decomposition is using a so-called measure of union information—which quantifies the information that is present in at least one of the sources—from which a synergy measure stems. In this paper, we introduce a new measure of union information based on adopting a communication channel perspective, compare it with existing measures, and study some of its properties. We also include a comprehensive critical review of characterizations of union information and synergy measures that have been proposed in the literature. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to implement a simulation model in @RISKTM and study the impact of incorporating random variables, such as the degree days in a traditional deterministic model, for calculating the optimum thickness of thermal insulation in walls. Currently, green buildings have become important because of the increasing worldwide interest in the reduction of environmental pollution. One method of saving energy is to use thermal insulation. The optimum thickness of these insulators has traditionally been calculated using deterministic models. With the information generated from real data using the degree days required in a certain zone in Palestine during winter, random samples of the degree days required annually in this town were generated for periods of 10, 20, 50, and 70 years. The results showed that the probability of exceeding the net present value of the cost calculated using deterministic analysis ranges from 0% to 100%, without regard to the inflation rate. The results also show that, for design lifetimes greater than 40 years, the risk of overspending is lower if the building lasts longer than the period for which it was designed. Moreover, this risk is transferred to whomever will pay the operating costs of heating the building. The contribution of this research is twofold: (a) a stochastic approach is incorporated into the traditional models that determine the optimum thickness of thermal insulation used in buildings, by introducing the variability of the degree days required in a given region; (b) a measure of the economic risk incurred by building heating is established as a function of the years of use for which the building is designed and the number of years it is actually used. [ABSTRACT FROM AUTHOR]

Considering the dynamic behavior of doubly‐fed induction generators (DFIGs) under the influence of random factors, this paper not only analyzes the phenomenon of stochastic instability and bifurcation of a DFIG dynamic variable in its random space when they are affected by environmental noise, but also proposes a method based on the Tkagi–Sugneo (T–S) fuzzy control strategy to control its stochastic bifurcation. First, a four‐dimensional stochastic dynamic DFIG model is established by using multiplicative white noise to simulate the influence of environmental noise on electrical variables, and stochastic central manifold theory is used to reduce the dimensionality of a planar model in the bifurcation neighborhood. Then, the stochastic stability of the model is investigated based on singular boundary theory, while the steady‐state probability density of the stochastic DFIG is determined using the Fokker–Plank–Kolmogorov equation to obtain the location and probability density of its stochastic P‐bifurcation. Finally, the influence of stochastic bifurcation behavior is eliminated by H∞‐fuzzy output feedback control. The numerical simulation results indicate that the location and probability of stochastic bifurcation in a DFIG will vary with the change in noise intensity, and the bifurcation parameter values and stochastic stability domain are obtained. The harm caused by random factors can be solved based on H∞‐fuzzy output feedback, which provides a theoretical basis for the stable operation of the DFIG system. [ABSTRACT FROM AUTHOR]

For decades, conventional wisdom maintained that binary 0–1 Bernoulli random variables cannot contain extra-binomial variation. Taking an unorthodox stance, Hilbe actively disagreed, especially for correlated observation instances, arguing that the universally adopted diagnostic Pearson or deviance dispersion statistics are insensitive to a variance anomaly in a binary context, and hence simply fail to detect it. However, having the intuition and insight to sense the existence of this departure from standard mathematical statistical theory, but being unable to effectively isolate it, he classified this particular over-/under-dispersion phenomenon as implicit. This paper explicitly exposes his hidden quantity by demonstrating that the variance in/deflation it represents occurs in an underlying predicted beta random variable whose real number values are rounded to their nearest integers to convert to a Bernoulli random variable, with this discretization masking any materialized extra-Bernoulli variation. In doing so, asymptotics linking the beta-binomial and Bernoulli distributions show another conventional wisdom misconception, namely a mislabeling substitution involving the quasi-Bernoulli random variable; this undeniably is not a quasi-likelihood situation. A public bell pepper disease dataset exhibiting conspicuous spatial autocorrelation furnishes empirical examples illustrating various features of this advocated proposition. [ABSTRACT FROM AUTHOR]

In reliability studies, we are interested in the behaviour of a system when it interacts with its surrounding environment. To assess the system's behaviour in a reliability sense, we can take the system's intrinsic quality as strength and the outcome of interactions as stress. Failure is observed whenever stress exceeds strength. Taking Y as a random variable representing the stress the system experiences and random variable X as its strength, the probability of not failing can be taken as a proxy for the reliability of the component and given as P (Y < X) = 1 − P (X < Y) . This way, in the present paper, it is considered that X and Y follow generalized extreme value distributions, which represent a family of continuous probability distributions that have been extensively applied in engineering and economic contexts. Our contribution deals with a more general scenario where stress and strength are not independent and copulas are used to model the dependence between the involved random variables. In such modelling framework, we explored the proper selection of copula models characterizing the dependence structure. The Gumbel–Hougaard, Frank, and Clayton copulas were used for modelling bivariate data sets. In each case, information criteria were considered to compare the modelling capabilities of each copula. Two economic applications, as well as an engineering one, on real data sets are discussed. Overall, an easy-to-use methodological framework is described, allowing practitioners to apply it to their own research projects. [ABSTRACT FROM AUTHOR]

In this paper, we investigate two types of the conditional strong law of large numbers with a new notion of conditionally independent random variables under G-expectation which are related to the symmetry G-function. Our limit theorem demonstrates that the cluster points of empirical averages fall within the bounds of the lower and upper conditional expectations with lower probability one. Moreover, for conditionally independent random variables with identical conditional distributions, we show the existence of two cluster points of empirical averages that correspond to the essential minimum and essential maximum expectations, respectively, with G-capacity one. [ABSTRACT FROM AUTHOR]

The authors investigate the complete moment convergence for arrays of rowwise independent random elements in Rademacher p Banach spaces. The results obtained in this paper improve the corresponding theorems of Hu et al. (Hu, T.-C., Rosalsky, A., Volodin, A., Zhang, S., 2021. A complete convergence theorem for row sums from arrays of rowwise independent random elements in Rademacher type p Banach spaces. II, Stochastic Anal. Appl., 39 (1), 177-193). Some corollaries and examples are also presented. [ABSTRACT FROM AUTHOR]

Localization performance in wireless networks has traditionally been benchmarked using the Cramér–Rao lower bound (CRLB), given a fixed geometry of anchor nodes and a target. However, by endowing the target and anchor locations with distributions, this paper recasts this traditional scalar benchmark as a random variable. The goal of this paper is to derive an analytical expression for the distribution of this now random CRLB, in the context of Time-of-Arrival-based positioning. To derive this distribution, this paper first analyzes how the CRLB is affected by the order statistics of the angles between consecutive participating anchors (i.e., internodal angles). This analysis reveals an intimate connection between the second largest internodal angle and the CRLB, which leads to an accurate approximation of the CRLB. Using this approximation, a closed-form expression for the distribution of the CRLB, conditioned on the number of participating anchors, is obtained. Next, this conditioning is eliminated to derive an analytical expression for the marginal CRLB distribution. Since this marginal distribution accounts for all target and anchor positions, across all numbers of participating anchors, it therefore statistically characterizes localization error throughout an entire wireless network. This paper concludes with a comprehensive analysis of this new network-wide-CRLB paradigm. [ABSTRACT FROM AUTHOR]

Blockchain technology brings about an opportunity to maintain decentralization in several applications, such as cryptocurrency. With the agents of a decentralized system operating independently, it calls for a consensus protocol that helps all nodes to agree on the state of the ledger. Most of the existing blockchains rely on Proof of Work (PoW) as the underlying consensus algorithm, resulting in a significant amount of electricity power consumption. Furthermore, it demands the miner to buy specific computation devices. Besides, a protocol to gather the society-related taxes such as public education funding and charities is lacking in existing consensus algorithms. In response, this paper proposes a new consensus algorithm, namely Proof of Humanity (PoH) aiming at gathering society-related taxes. According to PoH, the probability that an agent becomes a leader depends on its donations to non-profit accounts. Therefore, PoH encourages miners to donate money and gain mining power, its incentives, and transaction fees. The associated bureaucracy model is introduced briefly to address the required ecosystem for real case implementation of PoH. A distributed random variable generation algorithm is presented in this paper which ensures that the randomly selected leader is neither predictable nor adjustable. It is demonstrated that the proposed blockchain is totally robust against forking and possesses a high level of propagation speed, which ensures the scalability. Simulations show that the proposed blockchain network does not fail even in adverse scenarios where the majority of nodes refuse to propagate valid blocks. Besides, simulations reveal a suitable average block creation duration. [ABSTRACT FROM AUTHOR]

The histogram has all its bin widths equal to some non-random number arbitrary set by an analyst (EBWH). In the result, particular bin counts are random variables. This paper presents also a histogram that is constructed in a converse manner. Bin counts are all equal to some non-random number arbitrary set by an analyst (EBCH). In the result, particular bin widths are random variables. The first goal of the paper is a choose of constant bin width (of bin numbers k) in the EBWH, which maximize the similarity measure in the Monte Carlo simulation. The second goal is a choose of constant bin count in the EBCH, which maximize the similarity measure in the Monte Carlo simulation. The third goal is to present similarity measures between empirical and theoretical data. The fourth goal is the comparative analysis of two histogram methods by means of the frequency formula. The first additional goal is a tip how to proceed in EBCH when modulo(n,k)≠0. The second additional goal is the software in the form of a Mathcad file with the implementation of EBWH and EBCH. [ABSTRACT FROM AUTHOR]

The use of eddy covariance CO2 flux measurements in data assimilation and other applications requires an estimate of the random uncertainty. In previous studies, the two-tower approach has yielded robust uncertainty estimates, but care must be taken to meet the often competing requirements of statistical independence (non-overlapping footprints) and ecosystem homogeneity when choosing an appropriate tower distance. The role of the tower distance was investigated with help of a roving station separated between 8m and 34 km from a permanent EC grassland station. Random uncertainty was estimated for five separation distances with an extended two-tower approach which removed systematic differences of CO2 fluxes measured at two EC towers. This analysis was made for a dataset where (i) only similar weather conditions at the two sites were included and (ii) an unfiltered one. The extended approach, applied to weather-filtered data for separation distances of 95m and 173m gave uncertainty estimates in best correspondence with the independent reference method The introduced correction for systematic flux differences considerably reduced the overestimation of the two-tower based uncertainty of net CO2 flux measurements, e.g. caused by different environmental conditions at both EC towers. It is concluded that the extension of the two-tower approach can help to receive more reliable uncertainty estimates because systematic differences of measured CO2 fluxes which are not part of random error are filtered out. [ABSTRACT FROM AUTHOR]

We have studied the persistence probability p(t) of an active Brownian particle with shape asymmetry in two dimensions. The persistence probability is defined as the probability of a stochastic variable that has not changed its sign in a given fixed time interval. We have investigated two cases: (1) diffusion of a free active particle and (2) that of a harmonically trapped particle. In our earlier work, by Ghosh et al. [J. Chem. Phys. 152, 174901 (2020)], we had shown that p(t) can be used to determine the translational and rotational diffusion constant of an asymmetrically shaped particle. The method has the advantage that the measurement of the rotational motion of the anisotropic particle is not required. In this paper, we extend the study to an active anisotropic particle and show how the persistence probability of an anisotropic particle is modified in the presence of a propulsion velocity. Furthermore, we validate our analytical expression against the measured persistence probability from the numerical simulations of single particle Langevin dynamics and test whether the method proposed in our earlier work can help distinguish between active and passive anisotropic particles. [ABSTRACT FROM AUTHOR]

Surrogate models are frequently used to replace costly engineering simulations. A single surrogate is frequently chosen based on previous experience or by fitting multiple surrogates and selecting one based on mean cross-validation errors. A novel stacking strategy will be presented in this paper. This new strategy results from reinterpreting the model selection process based on the generalization error. For the first time, this problem is proposed to be translated into a well-studied financial problem: portfolio management and optimization. In short, it is demonstrated that the individual residues calculated by leave-one-out procedures are samples from a given random variable ϵi, whose second non-central moment is the i-th model's generalization error. Thus, a stacking methodology based solely on evaluating the behavior of the linear combination of the random variables ϵi is proposed. At first, several surrogate models are calibrated. The Directed Bubble Hierarchical Tree (DBHT) clustering algorithm is then used to determine which models are worth stacking. The stacking weights can be calculated using any financial approach to the portfolio optimization problem. This alternative understanding of the problem enables practitioners to use established financial methodologies to calculate the models' weights, significantly improving the ensemble of models' out-of-sample performance. A study case is carried out to demonstrate the applicability of the new methodology. Overall, a total of 124 models were trained using a specific dataset: 40 Machine Learning models and 84 Polynomial Chaos Expansion models (which considered 3 types of base random variables, 7 least square algorithms for fitting the up to fourth order expansion's coefficients). Among those, 99 models could be fitted without convergence and other numerical issues. The DBHT algorithm with Pearson correlation distance and generalization error similarity was able to select a subgroup of 23 models from the 99 fitted ones, implying a reduction of about 77% in the total number of models, representing a good filtering scheme which still preserves diversity. Finally, it has been demonstrated that the weights obtained by building a Hierarchical Risk Parity (HPR) portfolio perform better for various input random variables, indicating better out-of-sample performance. In this way, an economic stacking strategy has demonstrated its worth in improving the out-of-sample capabilities of stacked models, which illustrates how the new understanding of model stacking methodologies may be useful. [ABSTRACT FROM AUTHOR]

In the postmarket drug and vaccine safety surveillance, when the number of adverse events follows a Poisson distribution, the ratio between the exposed and the unexposed person‐time information is the random variable that governs the decision rule about the safety of the drug or vaccine. The probability distribution function of such a ratio is derived in this paper. Exact point and interval estimators for the relative risk are discussed as well as statistical hypothesis testing. To the best of our knowledge, this is the first paper that provides an unbiased estimator for the relative risk based on the person‐time ratio. The applicability of this new distribution is illustrated through a real data analysis aimed to detect increased risk of occurrence of Myocarditis/Pericarditis following mRNA COVID‐19 vaccination in Manitoba, Canada. [ABSTRACT FROM AUTHOR]

Obtaining wind energy for the production of electric energy plays a key role in overcoming the problems associated with climate change and the dwindling reserves of traditional types of energy resources. The purpose of this work is to analyze current methods of energy estimation and forecasting, to consider the main classifications of forecasts and methods used in their construction and to review the main types of mathematical distributions used to calculate the speed and power of wind flow, depending on specific geographical conditions. In recent years, there has been an increase in the capacity of modern wind generators, which has significantly improved the efficiency of wind energy parks. The initial stage in determining the feasibility of involving a particular energy source in the overall energy system of the region is a preliminary assessment of the energy potential, allowing one to determine the possible percentage of substitution of traditional energy. To solve such a problem, it is necessary to use models of energy supply. Evaluation of wind as a resource creates certain difficulties in modeling because this resource is stochastic and variable. In this regard, this paper proposes to consider various models for estimating wind energy potential, which can be classified into empirical models and models based on the application of modern intelligent data analysis technologies. The paper presents an analysis of the existing models for estimating the amount of energy, which can be used in a system designed to determine the most optimal configuration of the energy system based on the use of different conversion technologies most relevant to the case under study, and it also serves as the basis for creating digital twins designed to model and optimize the operation of the projected energy complex. [ABSTRACT FROM AUTHOR]

In this paper, we provide geometric insights with visualization into the multivariate Gaussian distribution and its entropy and mutual information. In order to develop the multivariate Gaussian distribution with entropy and mutual information, several significant methodologies are presented through the discussion, supported by illustrations, both technically and statistically. The paper examines broad measurements of structure for the Gaussian distributions, which show that they can be described in terms of the information theory between the given covariance matrix and correlated random variables (in terms of relative entropy). The content obtained allows readers to better perceive concepts, comprehend techniques, and properly execute software programs for future study on the topic's science and implementations. It also helps readers grasp the themes' fundamental concepts to study the application of multivariate sets of data in Gaussian distribution. The simulation results also convey the behavior of different elliptical interpretations based on the multivariate Gaussian distribution with entropy for real-world applications in our daily lives, including information coding, nonlinear signal detection, etc. Involving the relative entropy and mutual information as well as the potential correlated covariance analysis, a wide range of information is addressed, including basic application concerns as well as clinical diagnostics to detect the multi-disease effects. [ABSTRACT FROM AUTHOR]

In this paper, we study the complete f-moment convergence for arrays of rowwise m-negatively associated random variables under some general conditions. The results obtained in the paper extend and improve some previous known ones. As an application of the main results, we present the complete consistency for the estimator in a semiparametric regression model based on m-negatively associated errors. We perform some numerical simulations to verify the validity of the theoretical results based on finite samples. [ABSTRACT FROM AUTHOR]

In this paper, complete moment convergence for maximum of randomly weighted sums and complete convergence for randomly indexed sums of martingale difference sequences (MDS) are investigated under some proper and sufficient conditions. A Marcinkiewicz–Zygmund type strong law of large numbers (MZSLLN) for MDS is obtained. In addition, relationships among weights, weight functions and boundary functions are revealed in a sense. The results obtained in the paper generalize some corresponding ones for independent and some dependent random variables. As an application, strong consistency for estimators in a nonparametric regression model is established. [ABSTRACT FROM AUTHOR]

Context. The paper focuses on the problem of estimating the center of distribution of the random component of experimental data for density models with a negative kurtosis. Objective. The goal of this research is to develop methods to improve the efficiency of polynomial estimation of parameters of experimental data with a negative kurtosis coefficient. Method. The study applies a relatively new approach to obtaining estimates for the center of the probability distribution from the results of experimental data with a stochastic component. This approach is based on polynomial estimation methods that rely on the mathematical apparatus of Kunchenko's stochastic polynomials and the description of random variables by higher-order statistics (moments or cumulants). A number of probability density distributions with a negative kurtosis coefficient are used as models of the random component. As a measure of efficiency, the ratio of variance of the estimates for the center of the distribution found using polynomial and classical methods based on the parameter of amount of information obtained is used. The relative accuracy of polynomial estimates in comparison with the estimates of the mean, median and quantile estimates (center of curvature) is researched using the Monte Carlo method for multiple tests. Results. Polynomial methods for estimating the distribution center parameter for data models of probability distribution density with a negative kurtosis coefficient have been constructed. Conclusions. The research carried out in this paper confirms the potentially high efficiency of polynomial estimates of the coordinates of the center of the experimental data, which are adequately described by model distributions with a negative kurtosis. Statistical modeling has confirmed the effectiveness of the obtained estimates in comparison with the known non-parametric estimates based on the statistics of the mean, median, and quantile, even with small sample sizes. [ABSTRACT FROM AUTHOR]

A stochastic process {St}t∈[0,T] is called a martingale process if E[SnΙSm] = Sm almost surely for all m ≤ n. Martingale processes are often used in arbitrage-free pricing of financial assets due to its "fair game" property, that is, when an asset is modeled using martingales then no one can consistently make or lose money through trades in that asset. In this paper, we focus our attention on random variables that follow a Pareto distribution. We then look at the set of martingale measures on Rℝ² having marginal measures which are quantized using the so-called Un-quantization. We shall then represent the transition kernel of such martingale measures as bistochastic matrices. Finally, we shall give some characterization of the set of such matrices. [ABSTRACT FROM AUTHOR]