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This study proposes a new five-parameter distribution called the gamma-exponentiated Weibull–Poisson (GEWP) distribution. As an extension of the exponentiated Weibull–Poisson family, the GEWP distribution offers a more flexible tool for analyzing a wider variety of data due to its theoretically and practically advantageous properties. It encompasses established distributions like the exponential, Weibull, and exponentiated Weibull. The development of the GEWP distribution proposed in this paper is obtained by combining the gamma–exponentiated Weibull (GEW) and the exponentiated Weibull–Poisson (EWP) distributions. Therefore, it serves as an extension of both the GEW and EWP distributions. This makes the GEWP a viable alternative for describing the variability of occurrences, enabling analysis in situations where GEW and EWP may be limited. This paper analyzes the probability distribution functions and provides the survival and hazard rate functions, the sub-models, the moments, the quantiles, and the maximum likelihood estimation of the GEWP distribution. Then, the numerical experiments for the parameter estimation of GEWP distribution for some finite sample sizes are presented. Finally, the comparative study of GEWP distribution and its sub-models is investigated via the goodness of fit test with real datasets to illustrate its potentiality. [ABSTRACT FROM AUTHOR]

Economic globalization and the rapid development of the Internet make logistics systems more and more diversified, people and enterprises have greatly increased their requirements for logistics systems, and fourth party logistics has received more and more attention from people and related enterprises. In order to further study the routing problem under uncertain stochastic environments, this paper considers the fourth party logistics routing problem from a single manufacturer to multiple distributors with uncertain times and random supplies under the complete information symmetry scenario and symmetric transportation volume decision space. Then, an uncertain stochastic programming model is established with the minimum total cost as its core objective, and the total transportation time, manufacturer's supply, and distributor's demand as constraints. In order to solve the optimal path of the above problems, this paper transforms the uncertain stochastic programming model into a classical mathematical programming model based on the distribution functions of uncertain time and random supply. Finally, two numerical examples are given to verify the effectiveness of the proposed model. [ABSTRACT FROM AUTHOR]

In many manufacturing industries, the lifetime performance index C L is utilized to assess the manufacturing process performance for products following some lifetime distributions and subjecting them to progressive type I interval censoring. This paper aims to explore the sampling design required to achieve a specified level of significance and test power for products with lifetimes following the Exponentiated Frech'et distribution. Since lifetime distribution is an asymmetrical probability distribution, this investigation is related to the topic of asymmetrical probability distributions and applications in various fields. When the termination time is fixed but the number of intervals is variable, the optimal number of inspection intervals and sample sizes yielding the minimized total experimental costs are determined and tabulated. When the termination time is varying, the optimal number of inspection intervals, sample sizes, and equal interval lengths achieving the minimum total experimental costs are determined and tabulated. Optimal parameter values are displayed in tabular form for feasible applications for users. Additionally, a practical example is provided to illustrate how this sampling design can be used to collect data by using the optimal setup of parameters, followed by a testing procedure to assess the capability of the production process. [ABSTRACT FROM AUTHOR]

Limitations inherent to existing statistical distributions in capturing the complexities of real-world data often necessitate the development of novel models. This paper introduces the new exponential generalized inverse generalized Weibull (NEGIGW) distribution. The NEGIGW distribution boasts significant flexibility with symmetrical and asymmetrical shapes, allowing its hazard rate function to be adapted to many failure patterns observed in various fields such as medicine, biology, and engineering. Some statistical properties of the NEGIGW distribution, such as moments, quantile function, and Renyi entropy, are studied. Three methods are used for parameter estimation, including maximum likelihood, maximum product of spacing, and percentile methods. The performance of the estimation methods is evaluated via Monte Carlo simulations. The NEGIGW distribution excels in its ability to fit real-world data accurately. Five medical and engineering datasets are applied to demonstrate the superior fit of NEGIGW distribution compared to competing models. This compelling evidence suggests that the NEGIGW distribution is promising for lifetime data analysis and reliability assessments across different disciplines. [ABSTRACT FROM AUTHOR]

The aim of this paper is to consider the indeterminate Stieltjes moment problem together with all its probability density functions that have the positive real or the entire real axis as support. As a consequence of the concavity of the entropy function in both cases, there is one such density that has the largest entropy: we call it f hmax , the largest entropy density. We will prove that the Jaynes maximum entropy density (MaxEnt), constrained by an increasing number of integer moments, converges in entropy to the largest entropy density f hmax . Note that this kind of convergence implies convergence almost everywhere, with remarkable consequences in real applications in terms of the reliability of the results obtained by the MaxEnt approximation of the underlying unknown distribution, both for the determinate and the indeterminate case. [ABSTRACT FROM AUTHOR]

We study response time, a key performance characteristic of queueing mechanisms. The studied model incorporates both active and passive queue management, arbitrary service time distribution, as well as a complex model of arrivals. Therefore, the obtained formulas can be used to calculate the response time of many real queueing mechanisms with different features, by parameterizing adequately the general model considered here. The paper consists of two parts. In the first, mathematical part, we derive the distribution function for the response time, its density, and the mean value. This is done by constructing two systems of integral equations, for the distribution function and the mean value, respectively, and solving these systems with transform techniques. All the characteristics are derived both in the time-dependent and steady-state cases. In the second part, we present numerical values of the response time for a few system parameterizations and point out several of its properties, some rather counterintuitive. [ABSTRACT FROM AUTHOR]

In the following text, we comment on the main goals and results of these contributions: The first one, I Multistage Estimation of the Scale Parameter of Rayleigh Distribution with Simulation i [[1]], analyses the sequential estimation of the scale parameter of the Rayleigh distribution using the three-stage sequential sampling procedure developed by Hall [[15]]. The asymptotic behaviour of the proposed procedure is analysed in relation to point estimation and confidence interval estimation. In the second paper, I Multistage Estimation of the Rayleigh Distribution Variance i [[2]], the authors discuss the sequential multistage estimation of the variance of the Rayleigh distribution through the three-stage procedure developed by Hall [[15]]. The asymptotic distribution of the maximum likelihood estimator of the lifetime performance index is used to develop the testing procedure. [Extracted from the article]

This document is an editorial introducing a special issue of the journal "Symmetry" on the topic of symmetrical and asymmetrical distributions in statistics and data science. The editorial provides a summary of the contributions to the special issue, which include studies on topics such as control charts, copulas, statistical distributions, discretization methods, spatial dependence, queuing models, survival analysis, and reliability analysis. The authors invite researchers to submit contributions to a second volume of the special issue, with a potential focus on the hypergeometric and negative hypergeometric distributions. [Extracted from the article]

In this paper, we modify a semi-parameter estimation of the joint model for the mean medical cost function with time-dependent covariates to enable it to describe the nonlinear relationship between the longitudinal variable and time points by using polynomial approximation. The observation time points are discrete and not exactly the same for all subjects; in order to use all of the information, we first estimate the mean medical cost at the same observed time points for all subjects, and then we weigh these values using the kernel method. Therefore, a smooth mean function of medical costs can be obtained. The proposed estimating method can be used for asymmetric distribution statistics. The consistency of the estimator is demonstrated by theoretical analysis. For the simulation study, we first set up the values of parameters and non-parametric functions, and then we generated random samples for covariates and censored survival times. Finally, the longitudinal data of response variables could be produced based on the covariates and survival times. Then, numerical simulation experiments were conducted by using the proposed method and applying the JM package in R to the generated data. The estimated results for parameters and non-parametric functions were compared with different settings. Numerical results illustrate that the standard deviations of the parametric estimators decrease as the sample sizes increases and are much smaller than preassigned threshold value. The estimates of non-parametric functions in the model almost coincide with the true functions as shown in the figures of simulation study. We apply the proposed model to a real data set from a multicenter automatic defibrillator implantation trial. [ABSTRACT FROM AUTHOR]

In 1933, Kolmogorov published his book, I Foundations of the Theory of Probability i , laying the modern axiomatic foundations of probability theory and establishing his reputation as the world's leading expert in this field. Since the coefficients of the Taylor expansion of the CF are related to the integer moments of a random variable, we usually state that the probabilistic description of a random variable may also be given in terms of integer moments. The concept of the probability distribution and the random variables they describe underlies the mathematical discipline of probability theory and the science of statistics. [Extracted from the article]

Recently, the generalized type I progressive hybrid censoring scheme (GenT 1 PrHyCS) and generalized type II progressive hybrid censoring scheme (GenT 2 PrHyCS) have become quite popular in reliability studies. These two type censoring schemes are very complex due to the large number of parameters used to specify the censoring procedure. Therefore, in this paper, we consider a more general and more complex new censoring scheme. Also, we consider the exponential distribution(ExpD) and derive an expression for the density function of the MLE. We prove the exact distribution of the maximum likelihood estimator (MLE) and conditional moment generating function (CondMGF) of the MLE for the mean of the ExpD under a new censoring scheme. We then derive the exact confidence intervals (ConfItv) for the mean of the ExpD under a new censoring scheme. Finally, we present an example to explain the methods of inference derived for this paper. From the example data, it can be seen that PDF of MLE for ExpD under new censoring scheme is almost symmetrical. [ABSTRACT FROM AUTHOR]

This paper introduces a novel semicircular distribution obtained by applying the quadratic rank transmutation map to the stereographic semicircular exponential distribution, referred to as the transmuted stereographic semicircular exponential distribution (TSSCED). This newly proposed distribution exhibits enhanced flexibility compared to the baseline stereographic semicircular exponential distribution (SSCEXP). We conduct a comprehensive analysis of the model's properties and demonstrate its efficacy in data modeling through the application to a real dataset. [ABSTRACT FROM AUTHOR]

Recently, the uncertainty aspects of record values have been increasingly studied in the literature. In this paper, we study the residual Tsallis entropy of upper record values coming from random samples. In the continuous case, we define the Tsallis entropy quantity for the residual lifetime of upper record values in general distributions as the residual Tsallis entropy of upper record values coming from a uniform distribution. We also obtain a lower bound on the residual Tsallis entropy of upper data set values originating from an arbitrary continuous probability distribution. We also discuss the monotonic property of the residual Tsallis entropy of upper data sets. [ABSTRACT FROM AUTHOR]

In this work, we pose and solve, in tempered distribution spaces, an open problem proposed by Schrödinger in 1925. In particular, on the Schwartz distribution spaces, we define the linear continuous quantum operators associated with relativistic Hamiltonians of massive particles—particles with rest mass different from 0 and evolving in the four-dimensional Minkowski vector space M 4 . In other words, upon the tempered distribution state-space S ′ (M 4 , C) , we have found the most natural way to introduce the free-particle relativistic Hamiltonian operator and its corresponding Schrödinger equation (together with its conjugate equation, standing for antiparticles). We have found the entire solution space of our relativistic linear continuous evolution equation by completely solving a division problem in tempered distribution space. We define the Hamiltonian (Schwartz diagonalizable) operator as the principal square root of a strictly positive, Schwartz diagonalizable second-order differential operator (linked with the "Klein–Gordon operator" on the tempered distribution space S 4 ′ ). The principal square root of a Schwartz nondefective operator is defined in a straightforward way—following the heuristic fashion of some classic and greatly efficient quantum theoretical approach—in the paper itself. [ABSTRACT FROM AUTHOR]

Control charts are widely used in quality control and industrial sectors. Because of their important role, researchers are focusing on the development of new control charts. According to our study, there is no significant amount of published work on control charts using trigonometrically generated distribution methods. In this paper, we contribute to this interesting research gap by developing a new control chart using a sine-based distributional method. The proposed distributional method (or family of probability distributions) may be called a new modified sine-G family of distributions. Based on the new modified sine-G method, a novel modification of the Weibull distribution, namely, a new modified sine-Weibull distribution, is introduced. The new modified sine-Weibull distribution is flexible enough to capture symmetrical and asymmetrical behaviors of its density function. An industrial application is considered to show the importance and implacability of the proposed distribution in quality control. Based on the proposed model, an attribute control chart is developed under a truncated life test. The control chart limits (ARLs) are also computed for the proposed model. The ARLs of the proposed control chart are compared with the attribute control chart of the Weibull distribution. The results show that the developed chart is more efficient than the existing attribute control chart for the Weibull distribution. [ABSTRACT FROM AUTHOR]

To accurately model user preference information and ensure the symmetry or similarity between real user preference and the estimated value in product optimization design, an interactive estimation of a distribution algorithm integrated with surrogate-assisted fitness evaluation (SAF-IEDA) is proposed in this paper. Firstly, taking the evaluation information of a few individuals as training data, a similarity evaluation method between decision variables is proposed. Following that, a preference probability model is built to estimate the distribution probability of decision variables. Then, the preference utility function of individuals is defined based on the similarity of decision variables. Finally, the surrogate-assisted fitness evaluation is realized by optimizing the weight of the decision variables' similarities. The above strategies are incorporated into the interactive estimation of the distribution algorithm framework and applied to address the optimal product design problem and the indoor lighting optimization problem. The experimental results demonstrate that the proposed method outperforms the comparative method in terms of search efficiency and fitness prediction accuracy. [ABSTRACT FROM AUTHOR]

Probability life distributions usually describe the time to an event or survival time. Therefore, these life distributions play a crucial role in the analysis and projection of maximum life expectancy using the goodness of fit approach for nonparametric hypothesis testing. This study suggests a nonparametric technique to determine whether the data follow an exponential distribution or belong to the mathematical class of the moment generating function for used better than aged (UBAmgf). These tests can be applied to both censored and non-censored data. The upper percentile points of the test statistics are generated, and the suggested test's asymptotic normality is established. Some well-known alternative asymmetric probability models are used to compute the Pitman asymptotic relative efficiency (PARE) and powers of the proposed test. To demonstrate the paper's conclusions, some asymmetric real-world datasets are examined. [ABSTRACT FROM AUTHOR]

False data injection attack (FDIA) is a deliberate modification of measurement data collected by the power grid using vulnerabilities in power grid state estimation, resulting in erroneous judgments made by the power grid control center. As a symmetrical defense scheme, FDIA detection usually uses machine learning methods to detect attack samples. However, existing detection models for FDIA typically require large-scale training samples, which are difficult to obtain in practical scenarios, making it difficult for detection models to achieve effective detection performance. In light of this, this paper proposes a novel FDIA sample generation method to construct large-scale attack samples by introducing a hybrid Laplacian model capable of accurately fitting the distribution of data changes. First, we analyze the large-scale power system sensing measurement data and establish the data distribution model of symmetric Laplace distribution. Furthermore, a hybrid Laplace-domain symmetric distribution model with multi-dimensional component parameters is constructed, which can induce a deliberate deviation in the state estimation from its safe value by injecting into the power system measurement. Due to the influence of the multivariate parameters of the hybrid Laplace-domain distribution model, the sample deviation generated by this model can not only obtain an efficient attack effect, but also effectively avoid the recognition of the FDIA detection model. Extensive experiments are carried out over IEEE 14-bus and IEEE 118-bus test systems. The corresponding results unequivocally demonstrate that our proposed attack method can quickly construct large-scale FDIA attack samples and exhibit significantly higher resistance to detection by state-of-the-art detection models, while also offering superior concealment capabilities compared to traditional FDIA approaches. [ABSTRACT FROM AUTHOR]

This paper works on the claims data generated by individual policies which are randomly exposed to a period of continuous time. The main aim is to model the occurrence times of individual claims, as well as their developments given the feature information and exposure periods of individual policies, and thus project the outstanding liabilities. In this paper, we also propose a method to compute the moments of outstanding liabilities in an analytic form. It is significant for a general insurance company to more accurately project outstanding liabilities in risk management. It is well-known that the features of individual policies have effects on the occurrence of claims and their developments and thus the projection of outstanding liabilities. Neglecting the information can unquestionably decrease the prediction accuracy of stochastic reserving, where the accuracy is measured by the mean square error of prediction (MSEP), whose analytic form is computed according to the derived moments of outstanding liabilities. The parameters concerned in the proposed model are estimated based on likelihood and quasi-likelihood and the properties of estimated parameters are further studied. The asymptotic behavior of stochastic reserving is also investigated. The asymptotic distribution of parameter estimators is multivariate normal distribution which is a symmetric distribution and the asymptotic distribution of the deviation of the estimated loss reserving from theoretical loss reserve also follows a normal distribution. The confidence intervals for the parameter estimators and the deviation can be easily obtained through the symmetry of the normal distribution. Some simulations are conducted in order to support the main theoretical results. [ABSTRACT FROM AUTHOR]

It is extremely frequent for systems to fail in their demanding operating environments in many real-world contexts. When systems reach their lowest, highest, or both extreme operating conditions, they usually fail to perform their intended functions, which is something that researchers pay little attention to. The goal of this paper is to develop inference for multi-reliability using unit alpha power exponential distributions for stress–strength variables based on the progressive first failure. As a result, the problem of estimating the stress–strength function R, where X, Y, and Z come from three separate alpha power exponential distributions, is addressed in this paper. The conventional methods, such as maximum likelihood for point estimation, Bayesian and asymptotic confidence, boot-p, and boot-t methods for interval estimation, are also examined. Various confidence intervals have been obtained. Monte Carlo simulations and real-world application examples are used to evaluate and compare the performance of the various proposed estimators. [ABSTRACT FROM AUTHOR]

The class of skew normal distributions, introduced by Azzalini (1985), which is an asymmetric distribution and allows the presence of skewness. In this paper, we propose the pivotal quantity approach to construct the confidence interval for the mean, prediction interval for the mean of the future sample, and tolerance interval for the quantile. The fiducial distribution is also studied. Moreover, the performances of all the proposed confidence intervals are investigated through the Monte Carlo simulation. The pivotal quantity is a common method for calculating confidence intervals, which is used to construct confidence intervals in this paper. And the convergence of the obtained confidence interval is illustrated by the figures. Finally, a real data is used to explain proposed intervals in real life. [ABSTRACT FROM AUTHOR]

Under progressive type II censoring, the credible interval estimation and the credible region for parameters of two-parameter exponential distribution based on the Bayesian approach are presented in this paper. Two methods of Bayesian credible region are proposed under a given confidence level. We also presented the predictive interval of the future observation under this type of censoring. In order to compare the performance of our proposed Bayesian credible interval and region with the existing non-Bayesian methods, we conduct a simulation study by the Monte Carlo method to find the corresponding coverage probabilities. This research is related to the topic of asymmetrical probability distributions and applications across disciplines. Finally, one engineering example is used to demonstrate the Bayesian credible interval estimation methods proposed in this paper. [ABSTRACT FROM AUTHOR]

The paper considers the main ways of describing the process that characterizes the arrival of packets to a multiservice node of a telecommunications network. The features of the process under consideration are best represented by the cumulative distribution function A(t). It determines the distribution of the interval size between the moments of arrival of neighboring packets to the multiservice node. These intervals are random values. If it is not possible to perform measurements that allow the choosing of the A(t) function, then the distribution law of random variables is selected based on reasonable assumptions. For telephone switching nodes, the Poisson flow hypothesis was used, which is often similar to the symmetric distribution of the number of calls at time interval t. The results of traffic measurements for multiservice switching nodes showed that the studied distribution is inherently asymmetric. This paper mainly considers the possibility of choosing the A(t) function based on the measurement results presented in a form of the histogram a(t), which contains a series of values. This histogram allows us to obtain the desired distribution as a stepwise function by integration of the a(t). Practical interest is associated with the possibility of reducing the number of readings used to assess the A(t) function. The methods used by some authors are based on the application of arbitrarily chosen functions A(t) with so-called heavy tails. The proposed approach is based on real distributions defined at a finite time interval. As a result of this research, a methodology has been developed to accurately describe the process of packet arrival at the input of the multiservice node. The proposed methodology is based on analytical methods. It guarantees error minimization when investigating the probabilistic characteristics of a switching node in a multiservice network. [ABSTRACT FROM AUTHOR]

In this paper, a more flexible extension of the Fréchet distribution is introduced. The new distribution is defined by means of the stochastic representation as the quotient of two independent random variables, a Fréchet distribution and the power of a random variable, with uniform distribution in the interval (0, 1). We will call this new extension the slash Fréchet distribution and one of its main characteristics is that its tails are heavier than the Fréchet distribution. The general density of this distribution and some basic properties are determined. Its moments, skewness coefficients, and kurtosis are calculated. In addition, the estimation of the model parameters is obtained by the method of moments and maximum likelihood. Finally, three applications with real data are performed by fitting the new model and comparing it with the Fréchet distribution. [ABSTRACT FROM AUTHOR]

This paper investigates the concept of a Markovian queueing model with heterogeneous, intermittently available servers with feedback under a hybrid vacation policy. Both the asymmetric transition representation and the hybrid vacation policy are addressed in this article. The necessary and sufficient conditions for system stability are presented. In addition, the steady-state probability distribution of the queueing model was derived by employing the matrix geometric method. Furthermore, a few formulae were constructed to determine the model's performance indicators. Finally, the influence of system parameters was also investigated using some numerical examples. [ABSTRACT FROM AUTHOR]

This paper introduces extensions H 4 , p and X 8 , p of Horn's double hypergeometric function H 4 and Exton's triple hypergeometric function X 8 , taking into account recent extensions of Euler's beta function, hypergeometric function, and confluent hypergeometric function. Among the numerous extended hypergeometric functions, the primary rationale for choosing H 4 and X 8 is their comparable extension type. Next, we present various integral representations of Euler and Laplace types, Mellin and inverse Mellin transforms, Laguerre polynomial representations, transformation formulae, and a recurrence relation for the extended functions. In particular, we provide a generating function for X 8 , p and several bounding inequalities for H 4 , p and X 8 , p . We explore the utilization of the H 4 , p function within a probability distribution. Most special functions, such as the generalized hypergeometric function, the Beta function, and the p-extended Beta integral, exhibit natural symmetry. [ABSTRACT FROM AUTHOR]

In this paper, the estimation of the stress–strength reliability is taken into account when the stress and strength variables have unit Gompertz distributions with a similar scale parameter. The consideration of the unit Gompertz distribution in this context is because of its intriguing symmetric and asymmetric properties that can accommodate various histogram proportional-type data shapes. As the main contribution, the reliability estimate is determined via seven frequentist techniques using the ranked set sampling (RSS) and simple random sampling (SRS). The proposed methods are the maximum likelihood, least squares, weighted least squares, maximum product spacing, Cramér–von Mises, Anderson–Darling, and right tail Anderson–Darling methods. We perform a simulation work to evaluate the effectiveness of the recommended RSS-based estimates by using accuracy metrics. We draw the conclusion that the reliability estimates in the maximum product spacing approach have the lowest value compared to other approaches. In addition, we note that the RSS-based estimates are superior to those obtained by a comparable SRS approach. Additional results are obtained using two genuine data sets that reflect the survival periods of head and neck cancer patients. [ABSTRACT FROM AUTHOR]

In this study, kinetic interaction at the Cherenkov resonance between an electromagnetic wave pulse and a flow of electrons possessing a wide velocity spread at the scale of the characteristic range of the resonant electron wave interaction is considered. Due to the absence of a distribution function slope in the range of velocities corresponding to the electron wave's resonance, an electron's flow is a nearly stable media from the point of view of its interaction with a long enough wave pulse. In this paper, we explain our findings on the process of electron interaction with potential relief where the wave pulse is so short that the characteristic scale of the wave amplitude's inhomogeneity and the profile of the potential relief is comparable to the wavelength. We show that if an appropriate slippage between the phase and group velocities of the wave is provided, then the reflection process of particles from "fast" and "slow" close-to-resonance velocity fractions becomes non-symmetrical. This can provide a mechanism of amplification of short intensive wave pulses with electron flows with very large velocity spreads. [ABSTRACT FROM AUTHOR]

The use of information–theoretical methodologies to assess graph-based systems has received a significant amount of attention. Evaluating a graph's structural information content is a classic issue in fields such as cybernetics, pattern recognition, mathematical chemistry, and computational physics. Therefore, conventional methods for determining a graph's structural information content rely heavily on determining a specific partitioning of the vertex set to obtain a probability distribution. A network's entropy based on such a probability distribution is obtained from vertex partitioning. These entropies produce the numeric information about complexity and information processing which, as a consequence, increases the understanding of the network. In this paper, we study the Benes network and its novel-derived classes via different entropy measures, which are based on information functionals. We construct different partitions of vertices of the Benes network and its novel-derived classes to compute information functional dependent entropies. Further, we present the numerical applications of our findings in understanding network complexity. We also classify information functionals which describe the networks more appropriately and may be applied to other networks. [ABSTRACT FROM AUTHOR]

This paper discusses the triple sampling inference procedures for the mean of a symmetric distribution—the normal distribution when the coefficient of variation is known. We use the Searls' estimator as an initial estimate for the unknown population mean rather than the classical sample mean. In statistics literature, the normal distribution under investigation underlines almost all the natural phenomena with applications in many fields. First, we discuss the minimum risk point estimation problem under a squared error loss function with linear sampling cost. We obtained all asymptotic results that enhanced finding the second-order asymptotic risk and regret. Second, we construct a fixed-width confidence interval for the mean that satisfies at least a predetermined nominal value and find the second-order asymptotic coverage probability. Both estimation problems are performed under a unified optimal framework. The theoretical results reveal that the performance of the triple sampling procedure depends on the numerical value of the coefficient of variation—the smaller the coefficient of variation, the better the performance of the procedure. [ABSTRACT FROM AUTHOR]

The two-sided and one-sided empirical Bayes test (EBT) rules for the parameter of a generalized exponential distribution with contaminated data (errors in variables) are constructed by a deconvolution kernel method, respectively. Under the type of the supersmooth error distributions and the supersmooth errors with the error level can be controlled situations, the asymptotically optimal uniformly over a class of prior distributions and uniform rates of convergence of the corresponding regret for the proposed EBT rules are obtained with suitable conditions. The example study shows that the assumptions and conditions of the main results of this paper are satisfied easily by calculating. [ABSTRACT FROM AUTHOR]

Intuitionistic fuzzy sets (IFSs), as a representative variant of fuzzy sets, has substantial advantages in managing and modeling uncertain information, so it has been widely studied and applied. Nevertheless, how to perfectly measure the similarities or differences between IFSs is still an open question. The distance metric offers an elegant and desirable solution to such a question. Hence, in this paper, we propose a new distance measure, named D I F S , inspired by the Hellinger distance in probability distribution space. First, we provide the formal definition of the new distance measure of IFSs, and analyze the outstanding properties and axioms satisfied by D I F S , which means it can measure the difference between IFSs well. Besides, on the basis of D I F S , we further present a normalized distance measure of IFSs, denoted D I F S ˜ . Moreover, numerical examples verify that D I F S ˜ can obtain more reasonable and superior results. Finally, we further develop a new decision-making method on top of D I F S ˜ and evaluate its performance in two applications. [ABSTRACT FROM AUTHOR]

The use of statistical distributions to model life phenomena has received considerable attention in the literature. Recent studies have shown the potential of statistical distributions in modeling data in applied sciences, especially in environmental sciences. Among them, the Weibull distribution is one of the most well-known models that can be used very effectively for modeling data in the fields of pollution and gas emissions, to name a few. In this paper, we introduce a family of distributions, which we call the modified Alpha-Power Weibull-X family of distributions. Based on the proposed family, we introduce a new model with five parameters, the modified Alpha-Power Weibull–Weibull distribution. Some mathematical properties were determined. Bayesian and maximum likelihood estimates for the model parameters were derived. The MLEs, bootstrap and Bayesian HPD credibility intervals for the unknown parameters were performed. A Monte Carlo simulation study was performed to evaluate the performance of the estimates. A simulation study was performed based on the parameters of the proposed model. An application to the carbon dioxide emissions dataset was performed to predict unique symmetric and asymmetric patterns and illustrate the applicability and potential of the model. For this data set, the proposed model is compared with the modified alpha power Weibull exponential distribution and the two-parameter Weibull distribution. To show which of the competing distributions is the best, we draw on certain analytical tools such as the Kolmogorov–Smirnov test. Based on these analytical measures, we found that the new model outperforms the competing models. [ABSTRACT FROM AUTHOR]

In this paper, we will present a new, more flexible class of distributions with a domain in the interval (0,1), which presents heavier tails than other distributions in the same domain, such as the B e t a , Kumaraswamy, and Weibull Unitary distributions. This new distribution is obtained as a transformation of two independent random variables with a Weibull distribution, which we will call the Generalized Unitary Weibull distribution. Considering a particular case, we will obtain an alternative to the B e t a , Kumaraswamy, and Weibull Unitary distributions. We will call this new distribution of two parameters the type 2 unitary Weibull distribution. The probability density function, cumulative probability distribution, survival function, hazard rate, and some important properties that will allow us to infer are provided. We will carry out a simulation study using the maximum likelihood method and we will analyze the behavior of the parameter estimates. By way of illustration, real data will be used to show the flexibility of the new distribution by comparing it with other distributions that are known in the literature. Finally, we will show a quantile regression application, where it is shown how the proposed distribution fits better than other competing distributions for this type of application. [ABSTRACT FROM AUTHOR]

Abel's integral equation is an efficient singular integral equation that plays an important role in diverse fields of science. This paper aims to investigate Abel's integral equation and its solution using G α -transform, which is a symmetric relation between Laplace and Sumudu transforms. G α -transform, as defined via distribution space, is employed to establish a solution to Abel's integral equation, interpreted in the sense of distributions. As an application to the given theory, certain examples are given to demonstrate the efficiency and suitability of using the G α -transform method in solving integral equations. [ABSTRACT FROM AUTHOR]

The Weibull distribution, one of the most significant distributions with applications in numerous fields, is associated with numerous distributions such as generalized gamma distribution, exponential distribution, and Rayleigh distribution, which are asymmetric. Nevertheless, it shares a close relationship with a normal distribution where a process of transformation allows them to become symmetric. The Weibull distribution is commonly used to study the failure of components and phenomena. It has been applied to a variety of scenarios, including failure time, claims amount, unemployment duration, survival time, and especially wind speed data. A suitable area for installing a wind turbine requires a wind speed that is both sufficiently high and consistent, and so comparing the variation in wind speed in two areas is eminently desirable. In this paper, methods to estimate the confidence interval for the ratio of the coefficients of variation of two Weibull distributions are proposed and applied to compare the variation in wind speed in two areas. The methods are the generalized confidence interval (GCI), the method of variance estimates recovery (MOVER), and Bayesian methods based on the gamma and uniform priors. The Bayesian methods comprise the equal-tailed confidence interval and the highest posterior density (HPD) interval. The effectiveness of the methods was evaluated in terms of their coverage probabilities and expected lengths and also empirically applied to wind speed datasets from two different areas in Thailand. The results indicate that the HPD interval based on the uniform prior outperformed the others in most of the scenarios tested and so it is suggested for estimating the confidence interval for the ratio of the coefficients of variation of two Weibull distributions. [ABSTRACT FROM AUTHOR]

This article substantiates the relevance of the development of a cognitive model for assessing the security of information systems for various purposes, designed to support decision-making by officials of information security management bodies. The article analyzes scientific papers and research in this area, formulates the requirements for the functional capabilities of the model, and investigates and identifies the most appropriate modeling tools, based on the symmetry property that develops from integrated ontological and neuro-Bayesian models; typical clusters of information systems; tactics and techniques for the implementation of information security threats through the vulnerabilities of objects at various levels of the International Organization of Standardization/Open Systems Interconnection model (the ISO/OSI model); protective influences; and attacking influences. This approach allowed us to identify such objects of influence and their current vulnerabilities and scenarios for the implementation of information security threats; to calculate the joint probability distribution of information security events of various origins; and to simulate the process of operational management of information security. [ABSTRACT FROM AUTHOR]

The aim of this paper is to define and explore a certain class of analytic functions involving the (p , q) -Wanas operator related to the Janowski functions. We discuss geometric properties, growth and distortion bounds, necessary and sufficient conditions, the Fekete–Szegö problem, partial sums, and convex combinations for the newly defined class. We solve the Fekete–Szegö problem related to the convolution product and discuss applications to probability distribution. [ABSTRACT FROM AUTHOR]

We consider inference based on the profile likelihood function for the scale parameter of the generalized logistic distribution. This distribution is a generalization of the logistic distribution, a symmetric distribution like the normal distribution, and it has several applications in various fields. The generalization allows for possible left or right skewness, which makes it more flexible for modeling purposes. Inference procedures based on the profile likelihood of the scale parameter do not perform very well when the sample size is small, therefore, we derived adjustments to the profile likelihood for the generalized logistic distribution using results from higher-order likelihood theory. We obtained an adjustment based on the empirical covariances of certain scores of the profile likelihood function. Another adjustment is derived using ancillary statistics. The performance of the adjustments is investigated for point estimation of the scale parameter of the generalized logistic distribution using the bias and mean squared error criteria. Using an extensive simulation study, we found the adjustments are very successful in reducing the bias and the mean squared error of the maximum profile likelihood estimator in most situations. Moreover, we studied the performance of the profile likelihood ratio test and its adjustments using the criterion of the attainment of nominal sizes. We found that, when the sample size is small, the profile likelihood ratio test has empirical sizes that are highly inflated. Therefore, the test will be invalid in such situations. Simulation results show that the adjusted versions of the profile likelihood produce tests that attain the nominal sizes even for very small samples. This also applies to confidence intervals derived from these tests. In conclusion, both adjustments of the profile likelihood have significantly better performance than the unadjusted profile likelihood and are recommended, especially for small samples. In particular, the adjustment based on ancillary statistics appears to have the best overall performance in all situations considered. We applied the methods in this paper to real data on Carbon fibers. [ABSTRACT FROM AUTHOR]

A summary of the eleven papers published in this special issue is presented here. This volume was the last in a series of special issues dealing with symmetric and non-symmetric continuous probability distributions. The works presented in this issue propose new probabilistic models and extend the properties of other existing models in the statistical literature. [ABSTRACT FROM AUTHOR]

The weighted exponential distribution is a promising skewed distribution in the life-testing experiment. The joint progressive type-II censoring (JPC) scheme is an effective approach to reducing costs. In this paper, we consider the estimates of parameters of the weighted exponential distribution with the JPC data. Two populations, whose scale parameters are the same but the shape parameters of which are different, are chosen to be studied. We first evaluate the parameters with the maximum likelihood method. Because the maximum likelihood estimates of parameters cannot be obtained in closed form, we apply the Newton–Raphson method in this part. Bayesian estimates and the corresponding credible intervals under the squared error loss function are computed by using the Markov Chain Monte Carlo method. After that, we use the bootstrap method to calculate the associated confidence intervals of the unknown parameters. A simulation has been performed to test the feasibility of the above methods and real data analysis is also provided for illustrative purposes. [ABSTRACT FROM AUTHOR]

The paper presents the solution of the spatial transient problem of the impact of a moving heat flux source induced by the laser radiation on the surface of a half-space using the superposition principle and the method of transient functions. The hyperbolic equation of transient thermal conductivity accounting for the relaxation time is used to model the laser heating process. It is assumed that the heat flux is distributed symmetrically with respect to the center of the heating spot. The combined numerical and analytical algorithm has been developed and implemented, which allows one to determine the temperature distribution both on the surface and on the depth of the half-space. In this case, the principle of superposition is used with the use of a special symmetric Gaussian distribution to describe the model of a source of high-intensity heat flux. The use of such a symmetric distribution made it possible to calculate the integrals over the spatial variables analytically. The results of the work could be used to estimate the contribution of the conductive component in the overall heat transfer of materials exposed to intense heat flows (laser surface treatment, laser additive technologies, streamlining and heating of materials by high-enthalpy gases, etc.). [ABSTRACT FROM AUTHOR]

This paper studies the event-triggered adaptive tracking control problem of a class of pure-feedback systems. Via the backstepping method and the neural network approximation with the central symmetric distribution, an event-triggered adaptive neural network controller is designed. In particular, a dynamic gain driven by the tracking error is introduced into the event-triggering mechanism. Then, by using the Lyapunov stability theory, the boundedness of all the closed-loop signals is proved, and the tracking error falls into a prespecified ϵ -neighbourhood of zero. Meanwhile, the Zeno behaviour is avoided. Finally, two simulations verify the effectiveness of the proposed control scheme. [ABSTRACT FROM AUTHOR]

The research presented in this paper deals with analytic p-valent functions related to the generalized probability distribution in the open unit disk U. Using the Hadamard product or convolution, function f s (z) is defined as involving an analytic p-valent function and generalized distribution expressed in terms of analytic p-valent functions. Neighborhood properties for functions f s (z) are established. Further, by applying a previously introduced linear transformation to f s (z) and using an extended Libera integral operator, a new generalized Libera-type operator is defined. Moreover, using the same linear transformation, subclasses of starlike, convex, close-to-convex and spiralike functions are defined and investigated in order to obtain geometrical properties that characterize the new generalized Libera-type operator. Symmetry properties are due to the involvement of the Libera integral operator and convolution transform. [ABSTRACT FROM AUTHOR]

Using the data for upper record values, the interval estimation for the scale parameter of two-parameter exponential distribution is presented. In addition, two methods for the joint confidence region of two parameters are proposed. In terms of confidence region area, the simulation comparison of two methods of the confidence region is performed in this paper. The criterion of minimum confidence region area is used to obtain the optimal method of the confidence region. To illustrate our proposed interval estimation methods, one biometrical example is used and the corresponding confidence interval length and confidence region area are also calculated. Our research topic is related to the asymmetrical probability distributions and applications across disciplines. [ABSTRACT FROM AUTHOR]

In this paper, we construct a new Lagrangian discrete distribution, named the Lagrangian zero truncated Poisson distribution (LZTPD). It can be presented as a generalization of the zero truncated Poissson distribution (ZTPD) and an alternative to the intervened Poisson distribution (IPD), which was elaborated for modelling both over-dispersed and under-dispersed count datasets. The mathematical aspects of the LZTPD are thoroughly investigated, and its connection to other discrete distributions is crucially observed. Further, we define a finite mixture of LZTPDs and establish its identifiability condition along with some distributional aspects. Statistical work is then performed. The maximum likelihood and method of moment approaches are used to estimate the unknown parameters of the LZTPD. Simulation studies are also undertaken as an assessment of the long-term performance of the estimates. The significance of one additional parameter in the LZTPD is tested using a generalized likelihood ratio test. Moreover, we propose a new count regression model named the Lagrangian zero truncated Poisson regression model (LZTPRM) and its parameters are estimated by the maximum likelihood estimation method. Two real-world datasets are considered to demonstrate the LZTPD's real-world applicability, and healthcare data are analyzed to demonstrate the LZTPRM's superiority. [ABSTRACT FROM AUTHOR]

This paper studies three discretization methods to formulate discrete analogues of the well-known continuous generalized Pareto distribution. The generalized Pareto distribution provides a wide variety of probability spaces, which support threshold exceedances, and hence, it is suitable for modeling many failure time issues. Bayesian inference is applied to estimate the discrete models with different symmetric and asymmetric loss functions. The symmetric loss function being used is the squared error loss function, while the two asymmetric loss functions are the linear exponential and general entropy loss functions. A detailed simulation analysis was performed to compare the performance of the Bayesian estimation using the proposed loss functions. In addition, the applicability of the optimal discrete generalized Pareto distribution was compared with other discrete distributions. The comparison was based on different goodness-of-fit criteria. The results of the study reveal that the discretized generalized Pareto distribution is quite an attractive alternative to other discrete competitive distributions. [ABSTRACT FROM AUTHOR]

Coronavirus disease (COVID-19), which affects the whole world, continues to spread. This disease has infected and killed millions of people worldwide. To limit the rate of spread of the disease, early detection should be provided and then the infected person should be quarantined. This paper proposes a Deep Learning-based application for early and accurate diagnosis of COVID-19. Compared to other studies, this application's biggest difference and contribution are that it uses Tree Seed Algorithm (TSA)-optimized Artificial Neural Networks (ANN) to classify deep architectural features. Previous studies generally use fully connected layers for end-to-end learning classification. However, this study proves that even relatively simple AlexNet features can be classified more accurately with the TSA-ANN structure. The proposed hybrid model provides diagnosis with 98.54% accuracy for COVID-19 disease, which shows asymmetric distribution on Computed Tomography (CT) images. As a result, it is shown that using the proposed classification strategy, the features of end-to-end architectures can be classified more accurately. [ABSTRACT FROM AUTHOR]

Owing to the symmetry between drive–response systems, the discussions of synchronization performance are greatly significant while exploring the dynamics of neural network systems. This paper investigates the quasi-synchronization (QS) and quasi-uniform synchronization (QUS) issues between the drive–response systems on fractional-order variable-parameter neural networks (VPNNs) including probabilistic time-varying delays. The effects of system parameters, probability distributions and the order on QS and QUS are considered. By applying the Lyapunov–Krasovskii functional approach, Hölder's inequality and Jensen's inequality, the synchronization criteria of fractional-order VPNNs under controller designs with constant gain coefficients and time-varying gain coefficients are derived. The obtained criteria are related to the probability distributions and the order of the Caputo derivative, which can greatly avoid the situation in which the upper bound of an interval with time delay is too large yet the probability of occurrence is very small, and information such as the size of time delay and probability of occurrence is fully considered. Finally, two examples are presented to further confirm the effectiveness of the algebraic criteria under different probability distributions. [ABSTRACT FROM AUTHOR]

The motivation of this paper is to address the limitations of the conventional keypoint-based disparity estimation methods. Conventionally, disparity estimation is usually based on the local information of keypoints. However, keypoints may distribute sparsely in the smooth region, and keypoints with the same descriptors may appear in a symmetric pattern. Therefore, conventional keypoint-based disparity estimation methods may have limited performance in smooth and symmetric regions. The proposed algorithm is superpixel-based. Instead of performing keypoint matching, both keypoint and semiglobal information are applied to determine the disparity in the proposed algorithm. Since the local information of keypoints and the semi-global information of the superpixel are both applied, the accuracy of disparity estimation can be improved, especially for smooth and symmetric regions. Moreover, to address the non-uniform distribution problem of keypoints, a disparity refining mechanism based on the similarity and the distance of neighboring superpixels is applied to correct the disparity of the superpixel with no or few keypoints. The experiments show that the disparity map generated by the proposed algorithm has a lower matching error rate than that generated by other methods. [ABSTRACT FROM AUTHOR]