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Abstract The interplay of exchange rates among nations significantly influences both international and domestic trade, underscoring the pivotal role of the foreign exchange market (Forex) in a country's economic landscape. Forex fluctuations have a significant impact on the everyday lives of both government agencies and the public population, directly influencing a country's prosperity or misfortune. This work proposes an advanced fuzzy time series model that incorporates domain universe sub-partitioning, parameter adjustment optimization methodologies, and interval forecasting methods. We utilized this model to examine annual exchange rate patterns between the Pakistani rupee (PKR) and the US dollar (US$), comparing its forecast accuracy to that of other models. Our proposed methodology outperformed existing methodologies in terms of forecasting precision, providing stakeholders with valuable insights for making informed, data-driven business decisions that benefit both individual firms and the country overall.

It is essential to use compound families in statistical modeling for asymmetric-positive-skewed reliability and relief in real-life data for a number of reasons, including the capacity to capture underlying patterns and characteristics in the data more accurately. The current work introduces a novel compound and continuous G family of probability distributions characterized by two parameters, denoted as the compounded two-parameters general Rayleigh geometric family of probabilistic distributions. The research involves rigorous calculations to ascertain and analyze the relevant mathematical properties of this newly proposed family. We present aging notions under the new family, providing corresponding density, hazard, and survival functions. Notably, a specific emphasis is placed on the examination of the reciprocal-Weibull baseline model, both from a mathematical and statistical perspective, in a dedicated section of the study. To generate various multivariate and bivariate type distributions, the copula method is employed. These newly derived models hold significant utility for effectively modeling the multivariate and bivariate data. The study showcases the wide applicability in different sides under real-life datasets. The presented applications serve as concrete demonstrations of the family's effectiveness and versatility in addressing real-world scenarios.

Abstract Toxoplasma infection in humans is considered due to direct contact with infected cats. Toxoplasma infection (an endemic disease) has the potential to affect various organs and systems (brain, eyes, heart, lungs, liver, and lymph nodes). Bilinear incidence rate and constant population (birth rate is equal to death rate) are used in the literature to explain the dynamics of Toxoplasmosis disease transmission in humans and cats. The goal of this study is to consider the mathematical model of Toxoplasma disease with harmonic mean type incident rate and also consider that the population of humans and cats is not equal (birth rate and the death rate are not equal). In examining Toxoplasma transmission dynamics in humans and cats, harmonic mean incidence rates are better than bilinear incidence rates. The disease dynamics are first schematically illustrated, and then the law of mass action is applied to obtain nonlinear ordinary differential equations (ODEs). Analysis of the boundedness, positivity, and equilibrium points of the system has been analyzed. The reproduction number is calculated using the next-generation matrix technique. The stability of disease-free and endemic equilibrium are analyzed. Sensitivity analysis is also done for reproduction number. Numerical simulation shows that the infection is spread in the population when the contact rate $$\beta_{h}$$ β h and $$\beta_{c}$$ β c increases while the infection is reduced when the recovery rate $$\delta_{h}$$ δ h increases. This study investigates the impact of various optimal control strategies, such as vaccinations for the control of disease and the awareness of disease awareness, on the management of disease.

In recent years, several novel modeling applications have been able to better fit complex datasets, and they can provide insights that would not be possible with traditional distributions. A new model developed by mixing the exponential and gamma models, called the Garima model, is discussed in this article. This model exhibits a better behavior fit than exponential and Lindley, among others. When the reliability practitioner wishes to score a specific number of failed units, a generalized progressive-hybrid-censored Type-II technique has been proposed to reduce both the length and expense of a life test. Given the availability of the presented data, the difficulty of estimating the scale parameter and various reliability time aspects of the Garima model is investigated using likelihood and Bayes inferential approaches. In addition, when the Garima parameter is assumed to have gamma density prior, the Markovian-Chain via Monte-Carlo sampler from a symmetric loss is performed to obtain the symmetric Bayes’ infer. Besides the asymptotic confidence intervals, the highest intervals for all unknown subjects are also developed. Simulation comparisons are also carried out, and useful recommendations are provided. A real data application is examined based on genuine datasets from the physical sector to see how the examined approaches may be implemented in real-life situations.

The linear regression model is a widely used statistical tool that forms most modelling concepts' basis. The ordinary least square estimator is often adopted to estimate the model's parameters. The estimator is considered efficient when there are no violations of the classical regression assumptions. However, the estimator underperforms when the model violates the underlying assumptions of regression. One of the violated assumptions is the problem of multicollinearity. The problem occurs when there is a correlation among the model's independent variables. Many estimators have been proposed to solve this problem but the search for a better estimator continues. This study proposes an almost unbiased modified ridge-type (AUMRT) estimator which has proved to be comparatively superior to the existing ones. The performance of AUMRT was proven through theoretical proofs, simulations, and practical application to real-life data. The theoretical findings underscore the superiority of the proposed method, a notion reinforced by the outcomes of the simulation study. Specifically, the simulation results unequivocally demonstrate that, under specific conditions, the proposed estimator outperforms all other methods considered in this study. Moreover, validation through real-life application with Portland cement corroborates both the theoretical assertions and the simulation findings.

Nowadays, the increasing demands of smart cities require innovative technologies to optimize resource utilization and minimize costs. This research focuses on the construction sector, specifically the production of advanced high-strength steel. We address practical application problems from the iron and steel factory in Annaba, Algeria, known as fuzzy multi-objective linear programming problems. Two methods are applied: a mathematical model and Artificial Neural Networks (ANNs). Results indicate that the ANN approach offers significant time and effort savings.

This article covers the issue of evaluating the two shape parameters and reliability metrics of a novel Kumaraswamy-exponential lifetime distribution, whose density exhibits a left-skewed, right-skewed, or symmetric shape, through a type-II improved adaptive progressively censored sample. Both conventional and Bayesian viewpoints are used to evaluate the various parameters, which include point and interval estimations. While the estimation of one of the shape parameters requires a numerical solution, the other shape parameter estimation can be carried out in closed form by the classical method. Besides, the likelihood method’s asymptotic traits are employed to provide interval estimations for all parameters. Leveraging the Markov chain Monte Carlo process, the symmetric squared loss function and independent gamma priors are taken into account for calculating Bayes points and the highest posterior density interval estimations. To illustrate the accuracy, compare estimation methods, and show the applicability of the various suggested methods, a simulation examination and a pair of applications are looked at. In the end, four accuracy indicators are taken into consideration to figure out the best progressive censoring pattern. The numerical results indicate that when collecting samples using the suggested censored procedure, it is advisable to use the Bayesian estimation approach for evaluating the Kumaraswamy-exponential distribution.

Record statistics are generally used to measure a process’ stochastic behavior at previously unheard-of values, either upper or lower. It can be obtained or calculated using order statistics from a sample whose size is dictated by the values and order of occurrence of observations. Due to its finite support and bathtub-shaped hazard function, the Topp–Leone (TL) distribution is appealing for reliability investigations. In today’s practical applications, estimating unknown parameters and predicting upcoming records is an essential issue. Thus, in this work, the TL distribution’s lower record values are taken into account. Against the symmetric squared-error loss, both Bayes and empirical-Bayes approaches are used to estimate the TL’s parameter and calculate prediction limits for upcoming lower record values. A Monte Carlo simulation experiment is performed to assess the accuracy of the estimation methodologies provided. To demonstrate the operability and usefulness of the various approaches, two useful applications are studied. The numerical evaluations show that the suggested approaches perform effectively and that the provided estimations are adequate in practice.

Let K+(μi)={Qsiμi,si∈(m0μi,m+μi)}, i=1,2, be two CSK families generated by the nondegenerate probability measures μ1 and μ2 with support bounded from above. Define the set of measures L=K+(μ1)•K+(μ2)={Qs1μ1•Qs2μ2,s1∈(m0μ1,m+μ1)ands2∈(m0μ2,m+μ2)}, where Qs1μ1•Qs2μ2 denotes the Fermi convolution of Qs1μ1 and Qs2μ2. We prove that if L is still a CSK family (that is, L=K+(σ) for some nondegenerate probability measure ()σ), then the probability measures σ, μ1 and μ2 are of the free Poisson type and follow the free Poisson law up to affinity. The same result, regarding the free Poisson measure, is obtained if we consider the t-deformed free convolution t replacing the Fermi convolution • in the family of measures L.

In free probability, the theory of Cauchy–Stieltjes Kernel (CSK) families has recently been introduced. This theory is about a set of probability measures defined using the Cauchy kernel similarly to natural exponential families in classical probability that are defined by means of the exponential kernel. Within the context of CSK families, this article presents certain features of the Marchenko–Pastur law based on the Fermi convolution and the t-deformed free convolution. The Marchenko–Pastur law holds significant theoretical and practical implications in various fields, particularly in the analysis of random matrices and their applications in statistics, signal processing, and machine learning. In the specific context of CSK families, our study of the Marchenko–Pastur law is summarized as follows: Let K+(μ)={Qmμ(dx);m∈(m0μ,m+μ)} be the CSK family generated by a non-degenerate probability measure μ with support bounded from above. Denote by Qmμ•s the Fermi convolution power of order s>0 of the measure Qmμ. We prove that if Qmμ•s∈K+(μ), then μ is of the Marchenko–Pastur type law. The same result is obtained if we replace the Fermi convolution • with the t-deformed free convolution t.

We address the estimation of regression parameters for the ill-conditioned predictive linear model in this study. Traditional least squares methods often encounter challenges in yielding reliable results when there is multicollinearity. Therefore, we employ a better shrinkage method, average least squares-centered penalized regression (ALPR), as it offers a more efficient approach for handling multicollinearity than ridge regression. Additionally, we integrate ALPR with the principal component (PC) dimension reduction method for enhanced performance. We compared the proposed PCALPR estimation technique with existing ones for ill-conditioned problems through comprehensive simulations and real-life data analyses using the mean squared error. This integration results in superior model performance compared to other methods, highlighting the potential of combining dimensionality reduction techniques with penalized regression for enhanced model predictions.