Mathematical Properties Research Articles

Transmuted Power Gumbel Distribution: Estimation and Applications

In recent years, generalized distributions have been widely studied in statistics as they possess flexibility in applications. This is justified because the traditional distributions often do not provide good fit in relation to the real data set studied. This paper develops a Power Gumbel distribution using the quadratic rank transmutation map (QRTM). The new generalization is called the transmuted Power-Gumbel distribution. Various mathematical properties of this distribution including moments, moment generating function, quantile function, mean deviation and order statistics were also studied. These features support the legitimacy and robustness of the proposed distribution. The maximum likelihood method is used for estimating the model parameters, and the finite sample performance of the estimators are assessed by simulation studies indicating that their precision improves with larger sample sizes. The asymptotic confidence intervals for the parameters are also obtained based on asymptotic variance-covariance matrix. Finally, the usefulness of the proposed model is illustrated in an application to two real data sets and conclude that the four-parameter transmuted Power Gumbel distribution provides better fit than the other five models.

An innovative trigonometric technique for producing a new range of distributions: Implementations in the medical fields

Introducing new families of statistical distributions for data modeling in the medical sector is a fascinating area of research. Numerous methods have been developed and implemented for such purposes. Many of these methods rely on one or more additional parameters. However, the inclusion of new parameters can lead to certain consequences, such as re-parametrization problems. This paper also presents a novel family of distributions for generating statistical models. The proposed family is based on a trigonometric function and can be referred to as the weighted tangent family of distributions. A notable feature of the weighted tangent family is that it does not require any extra parameters. The paper also derives some mathematical properties and point estimators of the weighted tangent family. Furthermore, the paper focuses on a specific case within the weighted tangent family, namely the weighted tangent Weibull distribution. Finally, the paper considers three applications from the medical sector to demonstrate the effectiveness of the weighted tangent Weibull distribution. The practical illustrations underscore that the weighted tangent approach significantly improves the fitting capacity of the traditional models.

A new extension of XLindley distribution with mathematical properties, estimation, and application on the rainfall data

Data analysis is indeed quite popular and crucial in various fields such as meteorology, hydrology, epidemiology, economics, and biology. So, the purpose of this study is to introduce a new probability distribution to analyze rainfall data. A new extension of XLindley distribution is introduced using the alpha power transformation technique. The new distribution is named “alpha power transformed XLindley – (APTXL) distribution”. Mathematical properties of APTXL distribution are derived such as ordinary moments, moment generating function, quantile function, mean residual life function, and order statistics. The model parameters are estimated using five different estimation methods such as maximum likelihood, Anderson Darling, Cramer von Misses, Ordinary least squares, and weighted least squares. A comprehensive simulation study is utilized to check the behavior and efficiency of the derived estimators. It is found that the weighted least square approach efficiently estimates the model parameter than the other methods. This derivation of a new model not only contributes to theoretical advances in statistical methodology but also provides practical methods for rainfall data modeling. The APTXL distribution is used to evaluate rainfall datasets from Saudi Arabia. It is identified that the APTXL distribution provides efficient results as compared to considered competitive distributions.

Estimation and Some Statistical Properties of the hybrid Weibull Inverse Burr Type X Distribution with Application to Cancer Patient Data

In this paper, the Burr type X distribution is represented as a mixture with the hybrid Weibull family, called hybrid hybrid Hybrid Weibull Inverse Burr Type X (HWIBX) distribution in order to justify its usefulness in modeling statistical regularities in extreme values recorded in non-stationary streams of media events, the basic new distribution functions are presented in addition to several statistical and mathematical properties of the hybrid distribution as well as estimating the model parameters by three methods, namely maximum likelihood, least squares, and weighted least squares with Conduct Monte Carlo simulations for the three methods. Finally, these results are illustrated through an example of matching the hybrid distribution with data representing the values Bladder cancer and survival times (given in years) of a group comprising 46 patients treated with chemotherapy alone and knowing the results of the improvement of this expansion through comparison with some other distributions using some Statistical metrics.

A new family of generalized-G distributions with properties and applications

We introduce and study a new generalized family of distributions herein referred to as the Marshall–Olkin Exponentiated Half Logistic-Generalized-G (MO-EHL-GG). A generalized distribution is a broader class of probability distributions that includes various specific distributions. It has parameters that allow for flexibility in modeling different types of data. By combining the Marshall–Olkin generator, the exponentiated half logistic generator and the generalized generator, the MO-EHL-GG family of distributions is developed. The primary objective behind introducing this new distribution is its enhanced flexibility and the ability of its hazard rate function to exhibit diverse shapes, making it valuable for statistical analysis and modeling purposes. Special cases of the new model are presented. Mathematical and statistical properties of the distribution are investigated. Estimates of the parameters are provided and simulation studies are conducted to examine the consistency of the model’s estimates. The significance of the new model is finally investigated through applications to real-life data sets. Three datasets were analyzed, demonstrating superior performance of our proposed distribution compared to competing models with the same number of parameters.

Properties, estimation, and applications of the extended log-logistic distribution

This paper presents the exponentiated alpha-power log-logistic (EAPLL) distribution, which extends the log-logistic distribution. The EAPLL distribution emphasizes its suitability for survival data modeling by providing analytical simplicity and accommodating both monotone and non-monotone failure rates. We derive some of its mathematical properties and test eight estimation methods using an extensive simulation study. To determine the best estimation approach, we rank mean estimates, mean square errors, and average absolute biases on a partial and overall ranking. Furthermore, we use the EAPLL distribution to examine three real-life survival data sets, demonstrating its superior performance over competing log-logistic distributions. This study adds vital insights to survival analysis methodology and provides a solid framework for modeling various survival data scenarios.

Estimating Unknown Parameters and Disturbance Term in Uncertain Regression Models by the Principle of Least Squares

In the field of statistics, uncertain regression analysis occupies an important position. It can thoroughly analyze data sets contained in complex uncertainties, aiming to quantify and reveal the intricate relationships between variables. It is worth noting that the traditional least squares method only takes into account the reduction in the deviations between predictions and observations, and fails to fully consider the inherent characteristics of the correlation uncertainty distributions under the uncertain regression framework. In light of this, this paper constructs a statistical invariant with symmetric uncertainty distribution based on the observations and the disturbance term. It also proposes the least squares estimation of unknown parameters and disturbance term in the uncertain regression model based on the least squares principle and, combined with the mathematical properties of the normal uncertainty distribution, gives a numerical algorithm for solving specific estimates. Finally, in order to verify the effectiveness of the least squares estimation method proposed in this paper, we also design two numerical examples and an empirical study of forecasting of electrical power output.

Comparison of different estimation methods for the inverse power perk distribution with applications in engineering and actuarial sciences

Introducing the Inverse Power Perk distribution, this paper presents a versatile probability distribution designed to model positively skewed data with unprecedented flexibility. Building upon the Perk distribution, it accommodates a wide range of shapes including right-skewed, J-shaped, reversed J-shaped, and nearly symmetric densities, as well as hazard rates exhibiting various patterns of increase and decrease. The paper delves into the mathematical properties of this novel distribution and offers a comprehensive overview of estimation techniques, including maximum likelihood estimators, ordinary least square estimators, percentile-based estimators, maximum product of spacing estimators, Cramer-von Mises, weighted least squares estimators, and Anderson-Darling estimators. To assess the performance of these estimation methods across different sample sizes, Monte Carlo simulations are conducted. Through comparisons of average absolute error and mean squared error, the efficacy of each estimator is evaluated, shedding light on their suitability for both small and large samples. In a practical application, three real datasets, including insurance data, are employed to demonstrate the versatility of the current model, when comparing to existing alternatives. The IPP distribution offers significant advantages over traditional distributions, particularly in its superior ability to model tail risks, making it an invaluable tool for practitioners dealing with extreme values and rare events. Its computational efficiency further sets it apart, enabling more robust and faster analysis in large-scale datasets.This empirical analysis further underscores the utility and adaptability of the Inverse Power Perk model in capturing the nuances of diverse datasets, thereby offering valuable insights for practitioners in various fields.

Revisiting Weak Energy Condition and wormholes in Brans-Dicke gravity

It is known that the formation of a wormhole typically involves a violation of the Weak Energy Condition (WEC), but the reverse is not necessarily true. In the context of Brans-Dicke gravity, the generalized Campanelli-Lousto solution, which we shall unveil in this paper, demonstrates a WEC violation that coincides with the appearance of unbounded sheets of spacetime within the region where 0<r<rs. The emergence of a wormhole in the region where r>rs is thus only an indirect consequence of the WEC violation. Whereas the two regions, 0<r<rs and r>rs, in general are disconnected by physical singularities at r=rs, they are both part of the same mathematical solution, and their behavior can provide insights into the WEC, which is a mathematical property of the solution. Furthermore, we utilize the generalized Campanelli-Lousto solution to construct a Kruskal-Szekeres diagram, which exhibits a “gulf” sandwiched between the four quadrants in the diagram, a novel feature in Brans-Dicke gravity. Overall, our findings shed new light onto a complex interplay between the WEC and wormholes in the Brans-Dicke theory.

The Linear Algebra of the Pell-Lucas Matrix

In this paper, we introduce the Pell-Lucas and the symmetric Pell-Lucas matrices. The study delves into the linear algebra aspects of these matrices, analyzing their mathematical properties and relationships. We construct decompositions for both the Pell-Lucas matrix and its inverse matrix. We present the Cholesky factorization of the symmetric Pell-Lucas matrices. Furthermore, we derive some valuable identities and bounds for the eigenvalues of these symmetric matrices through the application of majorization notation.

Circular Clustering With Polar Coordinate Reconstruction.

There is a growing interest in characterizing circular data found in biological systems. Such data are wide-ranging and varied, from the signal phase in neural recordings to nucleotide sequences in round genomes. Traditional clustering algorithms are often inadequate due to their limited ability to distinguish differences in the periodic component θ. Current clustering schemes for polar coordinate systems have limitations, such as being only angle-focused or lacking generality. To overcome these limitations, we propose a new analysis framework that utilizes projections onto a cylindrical coordinate system to represent objects in a polar coordinate system optimally. Using the mathematical properties of circular data, we show that our approach always finds the correct clustering result within the reconstructed dataset, given sufficient periodic repetitions of the data. This framework is generally applicable and adaptable to most state-of-the-art clustering algorithms. We demonstrate on synthetic and real data that our method generates more appropriate and consistent clustering results than standard methods. In summary, our proposed analysis framework overcomes the limitations of existing polar coordinate-based clustering methods and provides an accurate and efficient way to cluster circular data.

End-to-End Signal Classification in Signed Cumulative Distribution Transform Space.

This paper presents a new end-to-end signal classification method using the signed cumulative distribution transform (SCDT). We adopt a transport generative model to define the classification problem. We then make use of mathematical properties of the SCDT to render the problem easier in transform domain, and solve for the class of an unknown sample using a nearest local subspace (NLS) search algorithm in SCDT domain. Experiments show that the proposed method provides high accuracy classification results while being computationally cheap, data efficient, and robust to out-of-distribution samples with respect to the existing end-to-end classification methods. The implementation of the proposed method in Python language is integrated as a part of the software package PyTransKit [1].

Application of multi-objective neural network algorithm in industrial polymerization reactors for reducing energy cost and maximising productivity

Optimization on an industrial scale is a complex task that involves fine-tuning the performance of large-scale systems and applications to make them more efficient and effective. This process can be challenging due to the increasing volume of work, growing system complexity, and the need to maintain optimal performance. Due to the significant power required for compression and the high costs of reactant materials, optimizing low-density polyethylene (LDPE) production to provide maximum productivity with a reduction of energy cost is required. However, it is not a simple process because the optimization problem of the LDPE tubular reactor consists of conflicting objective functions. Multi-objective neural network algorithm (MONNA) is a metaheuristic optimization method that provides a versatile and robust approach for solving complex, contradictory targets and diverse optimization problems that do not rely on specific mathematical properties of the problem. It is inspired by the structure and information-processing capabilities of biological neural networks. MONNA iteratively proposes solutions, evaluates its performance, and adjusts its approach based on feedback, which avoids complex mathematical formulations. In this work, we implement Multi-objective optimization neural network algorithm (MONNA) in LDPE tubular reactor for maximising productivity, conversion and minimising energy costs with three scenario of problem optimization, i.e. maximising productivity and reducing energy cost for the first problem (P1); increasing conversion and reducing energy costs for the second problem (P2); and increasing productivity and reducing by-products for the third problem (P3). The results show that the highest productivity, highest conversion, and lowest energy are 545.1 mil. RM/year, 0.314, and 0.672 mil. RM/year. The extreme points in the Pareto Front (PF) for various bi-objective situations provide practitioners with helpful information for selecting the best trade-off for the operational strategy. According to their preferences, decision-makers can use the resulting Pareto to decide on the most acceptable alternative. The decision variable plots show that both initiators in the reacting zone highly affected the optimal solution with the opposite action.

Exploring Quadrilaterals: An Interactive Task for 7th Grade Students Using GeoGebra Classroom

Using a Dynamic Geometry System (DGS) students can engage in a dynamic learning process that allows them to experiment, create strategies, make conjectures, argue, and deduce mathematical properties. A DGS enables the introduction of proofs, by providing visual aids. The proof of the conjectures made emerges as the next step towards formalising and understanding the contents covered and the introduction of automated deduction systems at schools can be an added value. The implementation of automated deduction systems in schools faces several challenges, such as curriculum-issues, teacher knowledge and the discrepancy between automated theorem provers and traditional practices of proving in schools. Synthetic provers based on a set of inference rules and forward chaining reasoning may provide a possible help to these challenges. With the aim of introducing formal proofs and inspiring students towards this goal, a geometric conjecture was chosen for 7th grade students to engage with, based on a rule set discussed in a previous work, “A rule based theorem prover: an introduction to proofs in secondary schools”. The study also aimed to showcase the application of a Geometry Automated Theorem Prover within the classroom setting. A sequence of tasks was created for students to complete using GeoGebra Classroom. In the first phase, they were asked to identify quadrilaterals, construct a parallelogram, and measure sides and angles until they formulated a conjecture, which was strengthened by the dynamic possibility of changing the constructed parallelogram. In the second phase, the students were provided with four rules and were asked to justify their results using these rules. By the end of these justifications, the students had a complete proof of the initial conjecture. A short video of the proof made using JGEx (Java Geometry Expert) was also shown to the students.

Mathematical model enhancing flash memory reliability through DFT-driven error correction coding

Flash memory, ubiquitous in diverse electronic devices, confronts persistent challenges stemming from inherent errors that jeopardize data integrity. This research situates itself at the intersection of these challenges and advancements, proposing an inventive error correction coding framework that harnesses the unique capabilities of analysis with a hybrid error control coding (HECC) approach. In the proposed work, a mathematical model aimed at enhancing the flash memory by identifying the error pattern within the pages using the discrete fourier transform (DFT). By incorporating distinctive DFT mathematical properties, the proposed technique intends to improve flash memory error correction beyond traditional methods. The flash storage defect detection and rectification results with hybrid error correction coding achieved bit error rate (BER) of 4.3e-6, latency 14.1, mean 15.1 and standard deviation 1.0. Error correction efficiency 98% and storage overhead 10%. With this approach results are significantly improving the error correction efficiency, reduce storage overhead and enhanced adaptability to diverse error patterns.

GeoGebra—A great platform for experiential learning, explorations and creativity in mathematics

In this paper we are presenting some examples of how GeoGebra is used in: a) explaining concepts of the first derivative, monotony, extremums; b) studying the properties of the function (strictly increasing/decreasing); c) demonstrating the mean value theorem. The results and the conclusions are based on the experiment carried out in the teaching and learning process in the chapter of derivatives in a third-year class of a secondary school in Albania. Also, there are some encouraging facts got by the use of GeoGebra: the double representation and the dynamic features of GeoGebra allow the students to quickly grasp the mathematical concepts and properties and be actively involved in further explorations. The use of GeoGebra tools is similar to the use of the tools of virtual games, and this is a great advantage to stimulate the students to learn mathematics and master their mathematical performance the same way they play games. On the other side, using GeoGebra, it is easier for the teachers to explain mathematical concepts, the properties of algebraic objects, to discuss about different situations and aspects of the subject under study and to methodically reason the results got. GeoGebra provides a very commodious environment for the students to effectively interoperate with each other. Our results showed that GeoGebra is effective in teaching and learning mathematics. GeoGebra software contributed in enhancing students’ understanding of mathematical concepts and improved students’ interest to learn mathematics. Also, we admit that not all the stages of implementing GeoGebra software in the classroom are flowing smoothly. Based on our experience and the other researchers, it is observed that the effectiveness is dependent on the way GeoGebra is integrated in the teaching and learning process, implying that the research must continue with the commitment of many researchers of mathematics, physics and other sciences.

A new two-parameter Rayleigh distribution: Statistical properties, actuarial measures, regression analysis, and applications

This paper presents a novel two-parameter distribution derived from the Rayleigh distribution, thoroughly investigating its essential mathematical properties. We employ estimation techniques to determine the proposed distribution's estimated parameters. Through extensive simulation studies, we analyze and evaluate the asymptotic behavior of the model estimators. Furthermore, we calculate various actuarial measures to highlight the practical utility of the proposed distribution in actuarial science. To further substantiate the applicability of our distribution, we perform a comprehensive regression analysis. The practical relevance of the proposed distribution is demonstrated by modeling a lifetime dataset from the insurance field, where it exhibits a superior fit compared to existing distributions. The findings suggest that the new distribution significantly improves modeling capabilities, making it a valuable tool for theoretical research and practical applications in fields requiring accurate lifetime data modeling.

Data-driven planning in socially responsible textile units amidst uncertainty

Social responsibility is a key factor for organizations to achieve sustainable success in the modern competitive market. This study proposes a hybrid VIKOR method to evaluate textile suppliers based on their social performance under uncertain and multi-objective conditions. The method can handle fuzzy, stochastic, and interval data simultaneously. The social criteria for the evaluation are derived from the literature review, the SA8000 standards, and the United Nations’ recommendations. Some of the criteria are also aligned with the World Bank’s Social Responsibility Diamond Model and the United Nations’ Sustainable Development Goals. Moreover, this study presents a fuzzy mathematical model for fabric purchasing that incorporates social criteria and the quality level into the optimization process. A goal programming method is developed based on the mathematical properties of the multi-objective model. A numerical study is conducted in the textile industry to demonstrate the efficiency and effectiveness of the proposed approaches. A comprehensive sensitivity analysis has been performed to investigate the behavior of the presented mathematical model under different conditions, and the results have been discussed concerning the insights for managers and stakeholders in the textile industry. The proposed model demonstrates that: 1) Customer demand and fabric orders have a direct relationship with increasing sales. 2) The fabric unit price has a direct impact on the quality value and requires cost control policies or pricing negotiations with suppliers. 3) Improving supplier and customer relations and formulating pricing consistent with social value are among the most important issues for the success of the textile and clothing industry. The best-fitting line successfully explains the variability of social performance and customer demand with an accuracy of 99.35 %.

To decompose the gender possibilities of the contemporary sestina: queer formalism reimagined through a poetic fixed form

ABSTRACT We are used to thinking of formal principles as either good or bad for gender expression. But might they be both, in some cases? This essay examines the recent work of queer formalism in narrative through the lens of a poetic verse form: the thirty-nine-line scheme of the sestina. Through this scheme, I show that formal principles contained in a single literary form can be conducive to the a priori antithetical notions of antinarrativity and queer relationality of form. On the one hand, the sestina exhibits a narrative-like metonymic extension through the repetition of end-words, a gesture that has been made queer through contemporary poems. On the other hand, the sestina inverts the linear temporality of narrative in its unique possibilities of anti-teleology through cyclic and non-linear stanzaic traversal, stemming from mathematical properties within abstract algebra. The duality produced in this single poetic form adds additional dimensions to queer formalist debates by splitting the problem of antinarrativity from the problem of formal constraint. I also show how Sara Ahmed’s Queer Phenomenology: Orientations, Objects, Others (2006) offers a reading of the orientations invoked by the sestina’s changing configurations of end-words.

A new two-parameter over-dispersed discrete distribution with mathematical properties, estimation, regression model and applications

This paper focuses on the derivation of a new two-parameter discrete probability distribution. The new model is derived by mixing Poisson and Loai distributions and is named “Poisson Loai Distribution”. The paper explores various mathematical properties of the new model, introducing a count-regression model based on this distribution. The parameters of the model are estimated using the maximum likelihood estimation method. A comprehensive simulation study is utilized to assess the behavior of derived estimators. The importance of the proposed distribution is confirmed through the analysis of three real datasets. It is found that the suggested distribution has the greatest match when compared to all rival distributions, and it may be a viable alternative for assessing dispersed count data.