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18 pages, 472 KiB

Open AccessArticle

Statistical Modeling Using a New Distribution with Application in Health Data

by Alanazi Talal Abdulrahman, Etaf Alshawarbeh and Mahmoud M. Abd El-Raouf

Mathematics 2023, 11(14), 3108; https://doi.org/10.3390/math11143108 - 14 Jul 2023

Viewed by 1595

The modeling of pandemics is significant in understanding and addressing the spread of infectious diseases. This study introduces a novel and highly flexible extension of the asymmetric unit Burr–Hatke distribution, termed the power Burr–Hatke distribution (PUBHD), and comprehensively investigates its mathematical properties. Multiple [...] Read more.

The modeling of pandemics is significant in understanding and addressing the spread of infectious diseases. This study introduces a novel and highly flexible extension of the asymmetric unit Burr–Hatke distribution, termed the power Burr–Hatke distribution (PUBHD), and comprehensively investigates its mathematical properties. Multiple parameter estimation methods are employed, and their asymptotic behavior is analyzed through simulation experiments. The different estimation techniques are compared to identify the most efficient approach for estimating the distribution’s parameters. To demonstrate the applicability and usefulness of the PUBHD model, we conducted a case study using a sample from the COVID-19 dataset and compared its performance with other established models. Our findings show that the PUBHD model provides a superior fit to the COVID-19 dataset and offers a valuable tool for accurately modeling real-life pandemics. Full article

(This article belongs to the Special Issue Distribution Theory and Application)

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28 pages, 1022 KiB

Open AccessArticle

Quality Control Testing with Experimental Practical Illustrations under the Modified Lindley Distribution Using Single, Double, and Multiple Acceptance Sampling Plans

by Yusra Tashkandy, Walid Emam, M. Masoom Ali, Haitham M. Yousof and Basma Ahmed

Mathematics 2023, 11(9), 2184; https://doi.org/10.3390/math11092184 - 05 May 2023

Cited by 4 | Viewed by 1091

Abstract

Quality control testing under acceptance sampling plans involves inspecting a representative sample of products or materials from a larger lot or batch to determine whether the lot meets predetermined quality standards. In this research, the modified Lindley distribution is used as a model [...] Read more.

Quality control testing under acceptance sampling plans involves inspecting a representative sample of products or materials from a larger lot or batch to determine whether the lot meets predetermined quality standards. In this research, the modified Lindley distribution is used as a model for lifetime study. When a life test is amputated at a pre-appropriated time to decide on the admission or refusal of the submitted batches, the problems of the single, double, and multiple (three and four stages) acceptance sampling strategies are introduced. The optimal sample sizes are computed for single, double, and multiple acceptance sampling plans to ensure that the veritable mean life is greater than the prescribed mean life at the stipulated consumer’s risk. The operating characteristic functions are investigated at diverse quality levels. For single, double, and multiple acceptance sampling plans, the minimal ratios of the veritable mean life to the prescribed mean life at the established percent of the producer’s risk are obtained. To demonstrate the uses of single, double, and multiple, some numerical experiments are presented. Full article

(This article belongs to the Special Issue Distribution Theory and Application)

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26 pages, 9086 KiB

Open AccessArticle

Risk Analysis and Estimation of a Bimodal Heavy-Tailed Burr XII Model in Insurance Data: Exploring Multiple Methods and Applications

by Haitham M. Yousof, S. I. Ansari, Yusra Tashkandy, Walid Emam, M. Masoom Ali, Mohamed Ibrahim and Salwa L. Alkhayyat

Mathematics 2023, 11(9), 2179; https://doi.org/10.3390/math11092179 - 05 May 2023

Cited by 5 | Viewed by 1117

Abstract

Actuarial risks can be analyzed using heavy-tailed distributions, which provide adequate risk assessment. Key risk indicators, such as value-at-risk, tailed-value-at-risk (conditional tail expectation), tailed-variance, tailed-mean-variance, and mean excess loss function, are commonly used to evaluate risk exposure levels. In this study, we analyze [...] Read more.

Actuarial risks can be analyzed using heavy-tailed distributions, which provide adequate risk assessment. Key risk indicators, such as value-at-risk, tailed-value-at-risk (conditional tail expectation), tailed-variance, tailed-mean-variance, and mean excess loss function, are commonly used to evaluate risk exposure levels. In this study, we analyze actuarial risks using these five indicators, calculated using four different estimation methods: maximum likelihood, ordinary least square, weighted least square, and Cramer-Von-Mises. To achieve our main goal, we introduce and study a new distribution. Monte Carlo simulations are used to assess the performance of all estimation methods. We provide two real-life datasets with two applications to compare the classical methods and demonstrate the importance of the proposed model, evaluated via the maximum likelihood method. Finally, we evaluate and analyze actuarial risks using the abovementioned methods and five actuarial indicators based on bimodal insurance claim payments data. Full article

(This article belongs to the Special Issue Distribution Theory and Application)

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27 pages, 418 KiB

Open AccessArticle

Construction of Column-Orthogonal Designs with Two-Dimensional Stratifications

by Song-Nan Liu, Min-Qian Liu and Jin-Yu Yang

Mathematics 2023, 11(6), 1549; https://doi.org/10.3390/math11061549 - 22 Mar 2023

Cited by 1 | Viewed by 1036

Abstract

For the design of computer experiments, column orthogonality and space-filling are two desirable properties. In this paper, we develop methods for constructing a new class of column-orthogonal designs (ODs) with two-dimensional stratifications on finer grids, including orthogonal Latin hypercube designs (OLHDs) as special [...] Read more.

For the design of computer experiments, column orthogonality and space-filling are two desirable properties. In this paper, we develop methods for constructing a new class of column-orthogonal designs (ODs) with two-dimensional stratifications on finer grids, including orthogonal Latin hypercube designs (OLHDs) as special cases. In addition to being column-orthogonal, these designs have good space-filling properties in two dimensions. The resulting designs achieve stratifications on

s^{2} \times s

or

s \times s^{2}

grids, and most column pairs satisfy stratifications on

s^{2} \times s^{2}

grids. Moreover, many column pairs can achieve stratifications on

s^{4} \times s^{2}

and

s^{2} \times s^{4}

grids. Furthermore, the obtained space-filling ODs can have

s^{6}

levels,

s^{4}

levels, and mixed levels, as required for different needs. Full article

(This article belongs to the Special Issue Distribution Theory and Application)

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15 pages, 334 KiB

Open AccessArticle

New Approaches on Parameter Estimation of the Gamma Distribution

by Xiao Ke, Sirao Wang, Min Zhou and Huajun Ye

Mathematics 2023, 11(4), 927; https://doi.org/10.3390/math11040927 - 12 Feb 2023

Cited by 3 | Viewed by 1843

Abstract

This paper discusses new approaches to parameter estimation of gamma distribution based on representative points. In the first part, the existence and uniqueness of gamma mean squared error representative points (MSE-RPs) are discussed theoretically. In the second part, by comparing three types of [...] Read more.

This paper discusses new approaches to parameter estimation of gamma distribution based on representative points. In the first part, the existence and uniqueness of gamma mean squared error representative points (MSE-RPs) are discussed theoretically. In the second part, by comparing three types of representative points, we show that gamma MSE-RPs perform well in parameter estimation and simulation. The last part proposes a new Harrel–Davis sample standardization technique. Simulation studies reveal that the standardized samples can be used to improve estimation performance or generate MSE-RPs. In addition, a real data analysis illustrates that the proposed technique yields efficient estimates for gamma parameters. Full article

(This article belongs to the Special Issue Distribution Theory and Application)

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21 pages, 882 KiB

Open AccessArticle

A New One-Parameter Distribution for Right Censored Bayesian and Non-Bayesian Distributional Validation under Various Estimation Methods

by Walid Emam, Yusra Tashkandy, Hafida Goual, Talhi Hamida, Aiachi Hiba, M. Masoom Ali, Haitham M. Yousof and Mohamed Ibrahim

Mathematics 2023, 11(4), 897; https://doi.org/10.3390/math11040897 - 10 Feb 2023

Cited by 8 | Viewed by 1168

Abstract

We propose a new extension of the exponential distribution for right censored Bayesian and non-Bayesian distributional validation. The parameter of the new distribution is estimated using several conventional methods, including the Bayesian method. The likelihood estimates and the Bayesian estimates are compared using [...] Read more.

We propose a new extension of the exponential distribution for right censored Bayesian and non-Bayesian distributional validation. The parameter of the new distribution is estimated using several conventional methods, including the Bayesian method. The likelihood estimates and the Bayesian estimates are compared using Pitman’s closeness criteria. The Bayesian estimators are derived using three loss functions: the extended quadratic, the Linex, and the entropy functions. Through simulated experiments, all the estimating approaches offered have been assessed. The censored maximum likelihood method and the Bayesian approach are compared using the BB algorithm. The development of the Nikulin–Rao–Robson statistic for the new model in the uncensored situation is thoroughly discussed with the aid of two applications and a simulation exercise. For the novel model under the censored condition, two applications and the derivation of the Bagdonavičius and Nikulin statistic are also described. Full article

(This article belongs to the Special Issue Distribution Theory and Application)

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20 pages, 367 KiB

Open AccessArticle

Design Efficiency of the Asymmetric Minimum Projection Uniform Designs

by Qiming Bai, Hongyi Li, Shixian Zhang and Jiezhong Tian

Mathematics 2023, 11(3), 765; https://doi.org/10.3390/math11030765 - 03 Feb 2023

Cited by 1 | Viewed by 987

Abstract

Highly efficient designs and uniform designs are widely applied in many fields because of their good properties. The purpose of this paper is to study the issue of design efficiency for asymmetric minimum projection uniform designs. Based on the centered

L_{2}

discrepancy, [...] Read more.

Highly efficient designs and uniform designs are widely applied in many fields because of their good properties. The purpose of this paper is to study the issue of design efficiency for asymmetric minimum projection uniform designs. Based on the centered

L_{2}

discrepancy, the uniformity of the designs with mixed levels is defined, which is used to measure the projection uniformity of the designs. The analytical relationship between the uniformity pattern and the design efficiency is established for mixed-level orthogonal arrays with a strength of two. Moreover, a tight lower bound of the uniformity pattern is presented. The research is relevant in the field of experimental design by providing a theoretical basis for constructing the minimum number of projection uniform designs with a high design efficiency under a certain condition. These conclusions are verified by some numerical examples, which illustrate the theoretical results obtained in this paper. Full article

(This article belongs to the Special Issue Distribution Theory and Application)

15 pages, 666 KiB

Open AccessArticle

A Conway–Maxwell–Poisson Type Generalization of Hypergeometric Distribution

by Sudip Roy, Ram C. Tripathi and Narayanaswamy Balakrishnan

Mathematics 2023, 11(3), 762; https://doi.org/10.3390/math11030762 - 02 Feb 2023

Viewed by 1354

Abstract

The hypergeometric distribution has gained its importance in practice as it pertains to sampling without replacement from a finite population. It has been used to estimate the population size of rare species in ecology, discrete failure rate in reliability, fraction defective in quality [...] Read more.

The hypergeometric distribution has gained its importance in practice as it pertains to sampling without replacement from a finite population. It has been used to estimate the population size of rare species in ecology, discrete failure rate in reliability, fraction defective in quality control, and the number of initial faults present in software coding. Recently, Borges et al. considered a COM type generalization of the binomial distribution, called COM–Poisson–Binomial (CMPB) and investigated many of its characteristics and some interesting applications. In the same spirit, we develop here a generalization of the hypergeometric distribution, called the COM–hypergeometric distribution. We discuss many of its characteristics such as the limiting forms, the over- and underdispersion, and the behavior of its failure rate. We write its probability-generating function (pgf) in the form of Kemp’s family of distributions when the newly introduced shape parameter is a positive integer. In this form, closed-form expressions are derived for its mean and variance. Finally, we develop statistical inference procedures for the model parameters and illustrate the results by extensive Monte Carlo simulations. Full article

(This article belongs to the Special Issue Distribution Theory and Application)

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12 pages, 294 KiB

Open AccessArticle

Triple Designs: A Closer Look from Indicator Function

by Zujun Ou, Minghui Zhang and Hongyi Li

Mathematics 2023, 11(3), 750; https://doi.org/10.3390/math11030750 - 02 Feb 2023

Viewed by 1565

Abstract

A method of tripling for a three-level design, which triples both the run size and the number of factors of the initial design, has been proposed for constructing a design that can accommodate a large number of factors by combining all possible level [...] Read more.

A method of tripling for a three-level design, which triples both the run size and the number of factors of the initial design, has been proposed for constructing a design that can accommodate a large number of factors by combining all possible level permutations of its initial design. Based on the link between the indicator functions of a triple design and its initial design, the close relationships between a triple design and its initial design are built from properties such as resolution and orthogonality. These theoretical results present a closer look at a triple design and provide a solid foundation for a design constructed using the tripling method, where the constructed designs have better properties, such as high resolution and orthogonality, and are recommended for application in high dimension topics of statistics or large-scale experiments. Full article

(This article belongs to the Special Issue Distribution Theory and Application)

19 pages, 3937 KiB

Open AccessArticle

Stress–Strength Modeling Using Median-Ranked Set Sampling: Estimation, Simulation, and Application

by Amal S. Hassan, Ibrahim M. Almanjahie, Amer Ibrahim Al-Omari, Loai Alzoubi and Heba Fathy Nagy

Mathematics 2023, 11(2), 318; https://doi.org/10.3390/math11020318 - 07 Jan 2023

Cited by 12 | Viewed by 1442

Abstract

In this study, we look at how to estimate stress–strength reliability models, R₁ = P (Y < X) and R₂ = P (Y < X), where the strength X and stress Y have the same distribution in [...] Read more.

In this study, we look at how to estimate stress–strength reliability models, R₁ = P (Y < X) and R₂ = P (Y < X), where the strength X and stress Y have the same distribution in the first model, R₁, and strength X and stress Z have different distributions in the second model, R₂. Based on the first model, the stress Y and strength X are assumed to have the Lomax distributions, whereas, in the second model, X and Z are assumed to have both the Lomax and inverse Lomax distributions, respectively. With the assumption that the variables in both models are independent, the median-ranked set sampling (MRSS) strategy is used to look at different possibilities. Using the maximum likelihood technique and an MRSS design, we derive the reliability estimators for both models when the strength and stress variables have a similar or dissimilar set size. The simulation study is used to verify the accuracy of various estimates. In most cases, the simulation results show that the reliability estimates for the second model are more efficient than those for the first model in the case of dissimilar set sizes. However, with identical set sizes, the reliability estimates for the first model are more efficient than the equivalent estimates for the second model. Medical data are used for further illustration, allowing the theoretical conclusions to be verified. Full article

(This article belongs to the Special Issue Distribution Theory and Application)

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17 pages, 352 KiB

Open AccessArticle

A Method for Augmenting Supersaturated Designs with Newly Added Factors

by Chun-Wei Zheng, Zong-Feng Qi, Qiao-Zhen Zhang and Min-Qian Liu

Mathematics 2023, 11(1), 60; https://doi.org/10.3390/math11010060 - 23 Dec 2022

Cited by 1 | Viewed by 1232

Abstract

Follow-up experimental designs are popularly used in industry. In practice, some important factors may be neglected for various reasons in the first-stage experiment and they need to be added in the next stage. In this paper, we propose a method for augmenting supersaturated [...] Read more.

Follow-up experimental designs are popularly used in industry. In practice, some important factors may be neglected for various reasons in the first-stage experiment and they need to be added in the next stage. In this paper, we propose a method for augmenting supersaturated designs with newly added factors and augmented levels using the Bayesian D-optimality criterion. In addition, we suggest using the integrated Bayesian D-optimal augmented design to plan the follow-up experiment when the newly added factors have been allowed to vary in an appropriate region. Examples and simulation results show that the augmented designs perform well in improving identified rates of latent factor effects. Full article

(This article belongs to the Special Issue Distribution Theory and Application)

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14 pages, 494 KiB

Open AccessArticle

Dependence Structure Analysis and Its Application in Human Microbiome

by Shilan Li, Jianxin Shi, Paul Albert and Hong-Bin Fang

Mathematics 2023, 11(1), 9; https://doi.org/10.3390/math11010009 - 20 Dec 2022

Viewed by 1237

Abstract

The human microbiome has been recently shown to be associated with disease risks and has important implications in risk stratification and precision medicine. Due to abundant taxa in the human body, microbiome data are high-dimensional and compositional. Dirichlet distributions and their generalization are [...] Read more.

The human microbiome has been recently shown to be associated with disease risks and has important implications in risk stratification and precision medicine. Due to abundant taxa in the human body, microbiome data are high-dimensional and compositional. Dirichlet distributions and their generalization are used to characterize the dependence structures of microbial data. Another existing method for fitting microbiome data employed Gaussian graphical model using the centered log-transformation (CLR). However, Dirichlet distributions are not able to infer networks or to estimate some extremely rare probabilities. On the other hand, it is hard to interpret the network analysis results using CLR. Furthermore, the data analysis showed that there is a lack of efficient multivariate distributions for fitting microbiome data, which results in inadequate statistical inferences. In this paper, we propose new multivariate distributions for modeling the dependence structures of the high dimensional and compositional microbiome data using inverse gamma distributions and copula techniques. The data analysis in the American gut project shows our proposed methods perform well. Full article

(This article belongs to the Special Issue Distribution Theory and Application)

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22 pages, 798 KiB

Open AccessArticle

Testing Multivariate Normality Based on F-Representative Points

by Sirao Wang, Jiajuan Liang, Min Zhou and Huajun Ye

Mathematics 2022, 10(22), 4300; https://doi.org/10.3390/math10224300 - 16 Nov 2022

Cited by 2 | Viewed by 1346

Abstract

The multivariate normal is a common assumption in many statistical models and methodologies for high-dimensional data analysis. The exploration of approaches to testing multivariate normality never stops. Due to the characteristics of the multivariate normal distribution, most approaches to testing multivariate normality show [...] Read more.

The multivariate normal is a common assumption in many statistical models and methodologies for high-dimensional data analysis. The exploration of approaches to testing multivariate normality never stops. Due to the characteristics of the multivariate normal distribution, most approaches to testing multivariate normality show more or less advantages in their power performance. These approaches can be classified into two types: multivariate and univariate. Using the multivariate normal characteristic by the Mahalanobis distance, we propose an approach to testing multivariate normality based on representative points of the simple univariate F-distribution and the traditional chi-square statistic. This approach provides a new way of improving the traditional chi-square test for goodness-of-fit. A limited Monte Carlo study shows a considerable power improvement of the representative-point-based chi-square test over the traditional one. An illustration of testing goodness-of-fit for three well-known datasets gives consistent results with those from classical methods. Full article

(This article belongs to the Special Issue Distribution Theory and Application)

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24 pages, 383 KiB

Open AccessArticle

Local Closure under Infinitely Divisible Distribution Roots and Esscher Transform

by Zhaolei Cui, Yuebao Wang and Hui Xu

Mathematics 2022, 10(21), 4128; https://doi.org/10.3390/math10214128 - 05 Nov 2022

Cited by 1 | Viewed by 836

Abstract

In this paper, we show that the local distribution class

L_{l o c} \cap {OS}_{l o c}

is not closed under infinitely divisible distribution roots, i.e., there is an infinitely divisible distribution which belongs to the class, while the corresponding Lévy [...] Read more.

In this paper, we show that the local distribution class

L_{l o c} \cap {OS}_{l o c}

is not closed under infinitely divisible distribution roots, i.e., there is an infinitely divisible distribution which belongs to the class, while the corresponding Lévy distribution does not. Conversely, we give a condition, under which, if an infinitely divisible distribution belongs to the class

L_{l o c} \cap {OS}_{l o c}

, then so does the Lévy distribution. Furthermore, we find some sufficient conditions that are more concise and intuitive. Using different methods, we also give a corresponding result for another local distribution class, which is larger than the above class. To prove the above results, we study the local closure under random convolution roots. In particular, we obtain a result on the local closure under the convolution root. In these studies, the Esscher transform of distribution plays a key role, which clarifies the relationship between these local distribution classes and related global distribution classes. Full article

(This article belongs to the Special Issue Distribution Theory and Application)

28 pages, 4357 KiB

Open AccessArticle

Representative Points from a Mixture of Two Normal Distributions

by Yinan Li, Kai-Tai Fang, Ping He and Heng Peng

Mathematics 2022, 10(21), 3952; https://doi.org/10.3390/math10213952 - 24 Oct 2022

Cited by 4 | Viewed by 1295

Abstract

In recent years, the mixture of two-component normal distributions (MixN) has attracted considerable interest due to its flexibility in capturing a variety of density shapes. In this paper, we investigate the problem of discretizing a MixN by a fixed number of points under [...] Read more.

In recent years, the mixture of two-component normal distributions (MixN) has attracted considerable interest due to its flexibility in capturing a variety of density shapes. In this paper, we investigate the problem of discretizing a MixN by a fixed number of points under the minimum mean squared error (MSE-RPs). Motivated by the Fang-He algorithm, we provide an effective computational procedure with high precision for generating numerical approximations of MSE-RPs from a MixN. We have explored the properties of the nonlinear system used to generate MSE-RPs and demonstrated the convergence of the procedure. In numerical studies, the proposed computation procedure is compared with the k-means algorithm. From an application perspective, MSE-RPs have potential advantages in statistical inference.Our numerical studies show that MSE-RPs can significantly improve Kernel density estimation. Full article

(This article belongs to the Special Issue Distribution Theory and Application)

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17 pages, 1293 KiB

Open AccessArticle

Quantitative Analysis of the Balance Property in Factorial Experimental Designs 2⁴ to 2⁸

by Ricardo Ramírez-Tapia, Armando Javier Ríos-Lira, Yaquelin Verenice Pantoja-Pacheco, José Antonio Vázquez-López and Edgar Augusto Ruelas-Santoyo

Mathematics 2022, 10(20), 3812; https://doi.org/10.3390/math10203812 - 15 Oct 2022

Viewed by 1260

Abstract

Experimental designs are built by using orthogonal balanced matrices. Balance is a desirable property that allows for the correct estimation of factorial effects and prevents the identity column from aliasing with factorial effects. Although the balance property is well known by most researchers, [...] Read more.

Experimental designs are built by using orthogonal balanced matrices. Balance is a desirable property that allows for the correct estimation of factorial effects and prevents the identity column from aliasing with factorial effects. Although the balance property is well known by most researchers, the adverse effects caused by the lack or balance have not been extensively studied or quantified. This research proposes to quantify the effect of the lack of balance on model term estimation errors: type I error, type II error, and type I and II error as well as R², R²_adj, and R²_pred statistics under four balance conditions and four noise conditions. The designs considered in this research include 2⁴–2⁸ factorial experiments. An algorithm was developed to unbalance these matrices while maintaining orthogonality for main effects, and the general balance metric was used to determine four balance levels. True models were generated, and a MATLAB program was developed; then a Monte Carlo simulation process was carried out. For each true model, 50,000 replications were performed, and percentages for model estimation errors and average values for statistics of interest were computed. Full article

(This article belongs to the Special Issue Distribution Theory and Application)

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12 pages, 327 KiB

Open AccessArticle

Huber Regression Analysis with a Semi-Supervised Method

by Yue Wang, Baobin Wang, Chaoquan Peng, Xuefeng Li and Hong Yin

Mathematics 2022, 10(20), 3734; https://doi.org/10.3390/math10203734 - 11 Oct 2022

Cited by 2 | Viewed by 1012

Abstract

In this paper, we study the regularized Huber regression algorithm in a reproducing kernel Hilbert space (RKHS), which is applicable to both fully supervised and semi-supervised learning schemes. Our focus in the work is two-fold: first, we provide the convergence properties of the [...] Read more.

In this paper, we study the regularized Huber regression algorithm in a reproducing kernel Hilbert space (RKHS), which is applicable to both fully supervised and semi-supervised learning schemes. Our focus in the work is two-fold: first, we provide the convergence properties of the algorithm with fully supervised data. We establish optimal convergence rates in the minimax sense when the regression function lies in RKHSs. Second, we improve the learning performance of the Huber regression algorithm by a semi-supervised method. We show that, with sufficient unlabeled data, the minimax optimal rates can be retained if the regression function is out of RKHSs. Full article

(This article belongs to the Special Issue Distribution Theory and Application)

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20 pages, 409 KiB

Open AccessArticle

Convergence of Uniformity Criteria and the Application in Numerical Integration

by Yang Huang and Yongdao Zhou

Mathematics 2022, 10(19), 3717; https://doi.org/10.3390/math10193717 - 10 Oct 2022

Cited by 2 | Viewed by 1180

Abstract

Quasi-Monte Carlo (QMC) methods have been successfully used for the estimation of numerical integrations arising in many applications. In most QMC methods, low-discrepancy sequences have been used, such as digital nets and lattice rules. In this paper, we derive the convergence rates of [...] Read more.

Quasi-Monte Carlo (QMC) methods have been successfully used for the estimation of numerical integrations arising in many applications. In most QMC methods, low-discrepancy sequences have been used, such as digital nets and lattice rules. In this paper, we derive the convergence rates of order of some improved discrepancies, such as centered

L_{2}

-discrepancy, wrap-around

L_{2}

-discrepancy, and mixture discrepancy, and propose a randomized QMC method based on a uniform design constructed by the mixture discrepancy and Baker’s transformation. Moreover, the numerical results show that the proposed method has better approximation than the Monte Carlo method and many other QMC methods, especially when the number of dimensions is less than 10. Full article

(This article belongs to the Special Issue Distribution Theory and Application)

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22 pages, 3665 KiB

Open AccessFeature PaperArticle

Methodology and Statistical Modeling of Social Capital Influence on Employees’ Individual Innovativeness in a Company

by Ekaterina V. Orlova

Mathematics 2022, 10(11), 1809; https://doi.org/10.3390/math10111809 - 25 May 2022

Cited by 5 | Viewed by 1653

Abstract

The research deals with the problem of identification and substantiation of mechanisms of social capital influence on individual innovativeness of employees, which increases the positive innovation effect in companies. The study proposes a new methodological approach and technology for assessing the social capital [...] Read more.

The research deals with the problem of identification and substantiation of mechanisms of social capital influence on individual innovativeness of employees, which increases the positive innovation effect in companies. The study proposes a new methodological approach and technology for assessing the social capital of employees, taking into account the factors of interpersonal and institutional trust, involvement in social networks, social norms, and its impact on the employee’s innovativeness. The methodology uses methods of system analysis and synthesis, expert assessments, statistical modeling, and survey. Numerical experiments are carried out using collected data from special surveys of employees of a machine-building company. An assessment of social capital and its impact on the employee’s innovativeness is determined and a statistically significant influence of the factors of “trust” and “social networks and connections” on social capital is set. It is revealed that the main determinant of innovativeness is the risk appetite. It is proved that the innovativeness model includes factors of “trust” and “social networks and connections”. The cumulative effect of social capital on innovativeness is positive and statistically significant. Full article

(This article belongs to the Special Issue Distribution Theory and Application)

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Review

Jump to: Research

25 pages, 572 KiB

Open AccessReview

A Review of Representative Points of Statistical Distributions and Their Applications

by Kai-Tai Fang and Jianxin Pan

Mathematics 2023, 11(13), 2930; https://doi.org/10.3390/math11132930 - 29 Jun 2023

Viewed by 1029

Abstract

Statistical modeling relies on a diverse range of statistical distributions, encompassing both univariate and multivariate distributions and/or discrete and continuous distributions. In the literature, numerous statistical methods have been proposed to approximate continuous distributions. The most commonly used approach is the use of [...] Read more.

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

(This article belongs to the Special Issue Distribution Theory and Application)

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