Search Results (61)

In recent years, there has been growing interest in discrete probability distributions due to their ability to model the complex behavior of real-world count data. In this paper, a new discrete mixture distribution based on two Weibull components is introduced, constructed using the general discretization approach. Several important statistical properties of the proposed distribution, including the survival function, hazard rate function, alternative hazard rate function, moments, quantile function, and order statistics are derived. It was concluded from the descriptive measures that the discrete mixture of two Weibull distributions transitions from being positively skewed with heavy tails to a more symmetric and light-tailed form. This demonstrates the high flexibility of the discrete mixture of two Weibull distributions in capturing a wide range of shapes as its parameter values vary. Estimation of the parameters is performed via maximum likelihood under Type II censoring scheme. A simulation study assesses the performance of the maximum likelihood estimators. Furthermore, the applicability of the proposed distribution is demonstrated using two real-life datasets. In summary, this paper constructs the discrete mixture of two Weibull distributions, investigates its statistical characteristics, and estimates its parameters, demonstrating its flexibility and practical applicability. These results highlight its potential as a powerful tool for modeling complex discrete data. Full article

Reliable estimation of low-flow statistics is essential for water quality regulation, ecological protection, and drought management. This study evaluates traditional univariate and two-component Mixture Probability Distributions for modeling 7-day annual minimum flows (7Q) using records from 293 gauging stations across Mexico’s 37 hydrological planning regions, each with at least 20 years of data. Candidate models include Lognormal-3, Gamma-3, Gumbel, Weibull-3, and mixtures (Gumbel–Gumbel, Gumbel–Weibull-3, Weibull-3–Gumbel, Weibull-3–Weibull-3). Parameters are estimated by maximum likelihood, goodness-of-fit is assessed with Kolmogorov–Smirnov and Anderson–Darling tests. Sampling uncertainty is quantified via nonparametric bootstrap, providing 95% confidence intervals for design return levels, including 7Q10. Mixture models are selected as the best fit at 253 of 293 stations (86.3%), with Weibull-3–Weibull-3 dominating (45.1% of all stations) followed by Gumbel–Weibull-3 and Weibull-3–Gumbel; univariate models account for only 13.7% of cases, mainly Lognormal-3, and Gumbel alone is never preferred. Gumbel-only and symmetric G–G mixtures yield negative low-flow return levels at some sites and are therefore considered physically implausible. In contrast, mixtures containing Weibull-3 components ensure non-negative support, provide superior fit to the lower tail, and generally produce narrower bootstrap confidence intervals than the best univariate alternatives, indicating more stable and defensible 7Q10 estimates and providing an additional criterion to distinguish between models with similar goodness-of-fit statistics. These findings have direct implications for Environmental Impact Assessment, effluent permitting, ecological flow setting, drought planning, and regional water policy. The results support integrating Weibull-based mixtures—especially Weibull-3–Weibull-3 and Gumbel–Weibull-3—into Mexico’s national framework for low-flow frequency analysis and regulatory design. Full article

This paper introduces the Lambert-Lindley distribution, a two-parameter extension of the Lindley model constructed through the Lambert-F generator. The new distribution retains the non-negative support of the Lindley distribution and provides additional flexibility by incorporating a shape parameter that controls skewness and tail behavior. Structural properties are derived, including the probability density function, cumulative distribution function, quantile function, hazard rate, and moments. Parameter estimation is addressed using the method of moments and maximum likelihood, and a Monte Carlo simulation study carried out to evaluate the performance of the proposed estimators. The practical applicability of the Lambert–Lindley distribution is demonstrated with two real datasets: stress rupture times of Kevlar/epoxy composites and hospital stay durations of breast cancer patients. Comparative analyses using goodness-of-fit tests and information criteria demonstrate that the proposed model can outperform classical alternatives such as the Gamma and Weibull distributions. Consequently, the Lambert–Lindley distribution emerges as a flexible alternative for modeling positive unimodal data in contexts such as material reliability studies and clinical duration analysis. Full article

This paper introduces a novel three-parameter probability model, the unit-Gompertz–Makeham (UGM) distribution, designed for modeling bounded data on the unit interval (0,1). By transforming the classical Gompertz–Makeham distribution, we derive a unit-support distribution that flexibly accommodates a wide range of shapes in both the density and hazard rate functions, including increasing, decreasing, bathtub, and inverted-bathtub forms. The UGM density exhibits rich patterns such as symmetric, unimodal, U-shaped, J-shaped, and uniform-like forms, enhancing its ability to fit real-world bounded data more effectively than many existing models. We provide a thorough mathematical treatment of the UGM distribution, deriving explicit expressions for its quantile function, mode, central and non-central moments, mean residual life, moment-generating function, and order statistics. To facilitate parameter estimation, eight classical techniques, including maximum likelihood, least squares, and Cramér–von Mises methods, are developed and compared via a detailed simulation study assessing their accuracy and robustness under varying sample sizes and parameter settings. The practical relevance and superior performance of the UGM distribution are demonstrated using two real-world engineering datasets, where it outperforms existing bounded models, such as beta, Kumaraswamy, unit-Weibull, unit-gamma, and unit-Birnbaum–Saunders. These results highlight the UGM distribution’s potential as a versatile and powerful tool for modeling bounded data in reliability engineering, quality control, and related fields. Full article

This study investigates date palm fiber (DPF) and sheep wool hybrid polyester composites with fiber loadings of 0%, 10%, 20%, and 30% by weight, fabricated by compression molding, to develop a sustainable and reliable material system. Experimental data from prior work were modeled using Weibull analysis for reliability evaluation and response surface methodology (RSM) for multi-objective optimization. Weibull statistics fitted a two-parameter distribution to tensile strength and fracture toughness, extracting shape (η) and scale ($\beta$) parameters to quantify variability and failure probability. The analysis showed that 20% hybrid content achieved the highest scale values ($\beta$ = 28.85 MPa for tensile strength and $\beta$ = 15.03 $\mathrm{M}\mathrm{P}\mathrm{a}\sqrt{\mathrm{m}}$ for fracture toughness) and comparatively low scatter (η = 10.39 and 9.2, respectively), indicating superior reliability. RSM quadratic models were developed for tensile strength, fracture toughness, thermal conductivity, acoustic attenuation, and water absorption, and were combined using desirability functions. The RSM optimization was found at 18.97% fiber content with a desirability index of 0.673, predicting 25.89 MPa tensile strength, 14.23 $\mathrm{M}\mathrm{P}\mathrm{a}\sqrt{\mathrm{m}}$ fracture toughness, 0.08 W/m·K thermal conductivity, 20.49 dB acoustic attenuation, and 5.11% water absorption. Overlaying Weibull cumulative distribution functions with RSM desirability surfaces linked probabilistic reliability zones (90–95% survival) to the deterministic optimization peak. This integration establishes a unified framework for designing natural fiber composites by embedding reliability into multi-property optimization. Full article

Accurate modelling of near surface wind speeds is essential for robust resource assessment, turbine design, and grid integration. This study presents a unified framework comparing four candidate marginal distributions—Weibull, Gamma, Lognormal, and Generalised Extreme Value (GEV)—across 21 years of daily observations from 11 sites in New South Wales and southern Queensland, Australia. Parameters are estimated by maximum likelihood, with L-moments used when numerical fitting fails. Univariate goodness-of-fit is evaluated via information criteria (Akaike Information Criterion, AIC; Bayesian Information Criterion, BIC) and distributional tests (Anderson–Darling, Cramér–von Mises, Kolmogorov–Smirnov). To capture spatial dependence, we fit an 11-dimensional regular vine (“R-vine”) copula to the probability-integral-transformed data, selecting pair-copula families by AIC and estimating parameters by sequential likelihood. A composite score (70% univariate, 30% copula) ranks distributions per location. Results demonstrate that Lognormal best matches central behaviour at most sites, Weibull remains competitive for bulk modelling, Gamma often excels in moderate tails, and GEV best represents extremes. All turbine yield results presented are illustrative, showing how statistical choices impact energy estimates; they should not be interpreted as operational forecasts. In a case study, 5000 joint simulations from the top-two models drive IEC V90 and E82 power curves, revealing up to 10% variability in annual energy yield due solely to marginal choice. This workflow provides a replicable template for comprehensive wind resource and load hazard analysis in complex terrains. Full article

This study evaluates six parameter estimation methods for the generalized extreme value (GEV) distribution: maximum likelihood estimation (MLE), two probability-weighted moments (PWM-UE and PWM-PP), and three robust two-stage order statistics estimators (TSOS-ME, TSOS-LMS, and TSOS-LTS). Their performance was assessed using simulation experiments under varying tail behaviors, represented by three types of GEV distributions: Weibull (short-tailed), Gumbel (light-tailed), and Fréchet (heavy-tailed) distributions, based on the mean squared error (MSE) and mean absolute percentage error (MAPE). The results showed that TSOS-LTS consistently achieved the lowest MSE and MAPE, indicating high robustness and forecasting accuracy, particularly for short-tailed distributions. Notably, PWM-PP performed well for the light-tailed distribution, providing accurate and efficient estimates in this specific setting. For heavy-tailed distributions, TSOS-LTS exhibited superior estimation accuracy, while PWM-PP showed a better predictive performance in terms of MAPE. The methods were further applied to real-world monthly maximum PM2.5 data from three air quality stations in Bangkok. TSOS-LTS again demonstrated superior performance, especially at Thon Buri station. This research highlights the importance of tailoring estimation techniques to the distribution’s tail behavior and supports the use of robust approaches for modeling environmental extremes. Full article

The Rayleigh distribution is a continuous probability distribution that is inherently asymmetric and commonly used to model right-skewed data. It holds significant importance across a wide range of scientific and engineering disciplines and exhibits structural relationships with several other asymmetric probability distributions, for example, Weibull and exponential distribution. This research proposes techniques for establishing credible intervals and confidence intervals for the single mean of the zero-inflated two-parameter Rayleigh distribution. The study introduces methods such as the percentile bootstrap, generalized confidence interval, standard confidence interval, approximate normal using the delta method, Bayesian credible interval, and Bayesian highest posterior density. The effectiveness of the proposed methods is assessed by evaluating coverage probability and expected length through Monte Carlo simulations. The results indicate that the Bayesian highest posterior density method outperforms the other approaches. Finally, the study applies the proposed methods to construct confidence intervals for the single mean using real-world data on COVID-19 total deaths in Singapore during October 2022. Full article

Vehicular Ad Hoc Networks (VANETs) serve as critical platforms for inter-vehicle communication within constrained ranges, facilitating information exchange. However, the inherent challenge of dynamic network topology poses persistent disruptions, hindering safety and emergency information exchange. An alternative generalised statistical model of the channel is proposed to capture the varying transmission range of the vehicle node. The generalised model framework uses simple wireless fading channel models (Weibull, Nakagami-m, Rayleigh, and lognormal) and the large vehicle obstructions to model the transmission range. This approach simplifies analysis of connection of vehicular nodes in environments were communication links are very unstable from obstructions from large vehicles and varying speeds. The connectivity probability is computed for two traffic models—free-flow and synchronized Gaussian unitary ensemble (GUE)—to simulate vehicle dynamics within a multi-lane road, enhancing the accuracy of VANET modeling. Results show that indeed the dynamic range distribution is impacted at shorter inter-vehicle distances and vehicle connectivity probability is lower with many obstructing vehicles. These findings offer valuable insights into the overall effects of parameters like path loss exponents and vehicle density on connectivity probability, thus providing knowledge on optimizing VANETs in diverse traffic scenarios. Full article

This study proposes a generalized family of distributions to enhance flexibility in modeling complex engineering and biomedical data. The framework unifies existing models and improves reliability analysis in both engineering and biomedical applications by capturing diverse system behaviors. We introduce a novel hybrid family of distributions that incorporates a flexible set of hybrid functions, enabling the extension of various existing distributions. Specifically, we present a three-parameter special member called the hybrid-Weibull–exponential (HWE) distribution. We derive several fundamental mathematical properties of this new family, including moments, random data generation processes, mean residual life (MRL) and its relationship with the failure rate function, and its related asymptotic behavior. Furthermore, we compute advanced information measures, such as extropy and cumulative residual entropy, and derive order statistics along with their asymptotic behaviors. Model identifiability is demonstrated numerically using the Kullback–Leibler divergence. Additionally, we perform a stress–strength (SS) reliability analysis of the HWE under two common scale parameters, supported by illustrative numerical evaluations. For parameter estimation, we adopt the maximum likelihood estimation (MLE) method in both density estimation and SS-parameter studies. The simulation results indicated that the MLE demonstrates consistency in both density and SS-parameter estimations, with the mean squared error of the MLEs decreasing as the sample size increases. Moreover, the average length of the confidence interval for the percentile and Student’s t-bootstrap for the SS-parameter becomes smaller with larger sample sizes, and the coverage probability progressively aligns with the nominal confidence level of 95%. To demonstrate the practical effectiveness of the hybrid family, we provide three real-world data applications in which the HWE distribution outperforms many existing Weibull-based models, as measured by AIC, BIC, CAIC, KS, Anderson–Darling, and Cramer–von Mises criteria. Furthermore, the HLW exhibits strong performance in SS-parameter analysis. Consequently, this hybrid family holds immense potential for modeling lifetime data and advancing reliability and survival analysis. Full article

To improve the quality of trench, backfill projects, this study utilizes solid waste to prepare a controllable low-strength material. Through uniaxial compression, three-point bending tests, and scanning electron microscopy (SEM), the mechanical performance evolution and fiber reinforcement mechanisms of the backfill material are revealed. Based on a two-parameter Weibull distribution probability model, an intrinsic correlation between the number of freeze–thaw cycles, damage variables, and compressive strength is established. The research results indicate that when the NaOH content is 3%, the water-to-solid ratio is 0.4, and the number of freeze–thaw cycles is 0, the sample’s mechanical properties reach their local optimum. After curing for 28 days, a significant amount of amorphous gel-like substance is formed inside the system, filling the intergranular spaces between aeolian sand particles, resulting in a relatively dense structure for the backfill material. In response to the degradation caused by the initial defects in the sample, fibers effectively prevent crack initiation. Based on the stochastic characteristics of freeze–thaw damage, the number of freeze–thaw cycles (n) follows the Weibull distribution model well. Using experimental data, evolution equations for the number of freeze–thaw cycles, intrinsic damage, and compressive strength were developed, ultimately establishing the intrinsic relationship between sample damage and strength. The findings provide theoretical support for addressing trench backfill engineering disasters in seasonally frozen regions. Full article

This study introduces two bivariate extensions of the recently proposed weighted discretized Fréchet–Weibull distribution, termed as bivariate weighted discretized Fréchet–Weibull (BWDFW) distributions. These models are specifically designed for analyzing two-dimensional discrete datasets and are developed using two distinct structural approaches: the minimum operator (BWDFW-I) and the maximum operator (BWDFW-II). A rigorous mathematical formulation is presented, encompassing the joint cumulative distribution function, joint probability mass function, and joint (reversed) hazard rate function. The dependence structure of the models is investigated, demonstrating their capability to capture positive quadrant dependence. Additionally, key statistical measures, including covariance, Pearson’s correlation coefficient, Spearman’s rho, and Kendall’s tau, are derived using the joint probability-generating function. For robust statistical inferences, the parameters of the proposed models are estimated via the maximum likelihood estimation method, with extensive simulation studies conducted to assess the efficiency and accuracy of the estimators. The practical applicability of the BWDFW distributions is demonstrated through their implementation in two real-world datasets: one from the aviation sector and the other from the security and safety domain. Comparative analyses against four existing discrete bivariate Weibull extensions reveal the superior performance of the BWDFW models, with BWDFW-I (minimum operator based) exhibiting greater flexibility and predictive accuracy than BWDFW-II (maximum operator based). These findings underscore the potential of the BWDFW models as effective tools for modeling and analyzing bivariate discrete data in diverse applied contexts. Full article

Using construction and demolition (C&D) waste in concrete production is a promising step toward environmental resilience amid the construction industry’s ecological footprint. The extensive history of using bricks in the construction of buildings has resulted in a considerable amount of waste associated with this commonly used material. This study aimed to assess the quality of concrete by examining the effect of replacing cement with varying percentages of recycled brick powder (RBP—0% to 50%). The primary objectives include evaluating the mechanical properties of concrete and establishing the feasibility of using RBP as a partial cement substitute. The investigation of target concrete can be divided into two phases: (i) laboratory investigation, and (ii) numerical investigation. In the laboratory phase, the performance of concrete with RBP was assessed under short-term dynamic and various static loads. The drop-weight test recommended by the ACI 544 committee was used to assess the short-term dynamic behavior (352 concrete discs). Furthermore, the behavior under static load was analyzed through compressive, flexural, and tensile strength tests. During the numerical phase, artificial neural network models (ANN) and fuzzy logic models (FL) were used to predict the results of 28-day compressive strength. The impact life with different failure probabilities was predicted based on the impact resistance results, by combining the Weibull distribution model. Additionally, an impact damage evolution equation was presented for mixtures containing RBP. The results show that the use of RBP up to 15% caused a slight decrease in compressive, flexural, and tensile strength (about 3–5%). Also, by replacing RBP up to 15%, the first crack strength decreased by 7.15% and the failure strength decreased by 6.46%. The average error for predicting 28-day compressive strength by FL and ANN models was recorded as 4.66% and 0.87%, respectively. In addition, the results indicate that the impact data follow the two-parameter Weibull distribution, and the R² value for different mixtures was higher than 0.9275. The findings suggest that incorporating RBP in concrete can contribute to sustainable construction practices by reducing the reliance on cement and utilizing waste materials. This approach not only addresses environmental concerns but also enhances the quality assessment of concrete, offering potential cost savings and resource efficiency for the construction industry. Real-world applications include using RBP-enhanced concrete in non-structural elements, such as pavements, walkways, and landscaping features, where high strength is not the primary requirement. Full article

This paper introduces a novel four-parameter discrete bivariate distribution, termed the bivariate discretized Fréchet–Weibull distribution (BDFWD), with marginals derived from the discretized Fréchet–Weibull distribution. Several statistical and reliability properties are thoroughly examined, including the joint cumulative distribution function, joint probability mass function, joint survival function, bivariate hazard rate function, and bivariate reversed hazard rate function, all presented in straightforward forms. Additionally, properties such as moments and their related concepts, the stress–strength model, total positivity of order 2, positive quadrant dependence, and the median are examined. The BDFWD is capable of modeling asymmetric dispersion data across various forms of hazard rate shapes and kurtosis. Following the introduction of the mathematical and statistical frameworks of the BDFWD, the maximum likelihood estimation approach is employed to estimate the model parameters. A simulation study is also conducted to investigate the behavior of the generated estimators. To demonstrate the capability and flexibility of the BDFWD, three distinct datasets are analyzed from various fields, including football score records, recurrence times to infection for kidney dialysis patients, and student marks from two internal examination statistical papers. The study confirms that the BDFWD outperforms competitive distributions in terms of efficiency across various discrete data applications. Full article

To enhance the impact resistance of high-performance concrete (HPC), a novel efficient solution was adopted by incorporating basalt fibers (BF) and polypropylene fibers (PF) as reinforcement materials in this work. To this end, the effects of single BF (BHC) and PF (PHC) as well as their combinations (BPHPC) on the impact energy consumption, ductility ratio, and toughness factor were explored through drop weight impact test of concrete considering fiber volume contents (0.1%, 0.15%, 0.2%) to evaluate the impact resistance of the concrete. The Weibull distribution function model is used to fit the drop weight impact test results and predict the probability of failure. Moreover, the fracture-resistance enhancement mechanism of fiber is analyzed at a microscopic level. Test results showed that the number of impacts resisted by the HPC can follow well the two-parameter Weibull distribution. Compared with the single BF and single PF, the combination of 0.15% BF and 0.1% PF yields favorable impact resistance, thus exhibiting a positive hybrid effect. Full article