Search Results (87)

The first (main) aim of the article is to define and practically apply the parameterized Kolmogorov–Smirnov (PKS) goodness-of-fit test for normality. The second contribution is to expand the family of empirical distribution functions with four new proposals. The third contribution is to construct a family of alternative distributions that includes both older and newer distributions. The fourth contribution is to calculate the power of the analyzed tests under alternative distributions with parameters chosen to be similar to the normal distribution in various ways. The new proposal is distinguished for left-skewed alternative distributions and symmetric distributions characterizing by negative excess kurtosis. The effectiveness of the PKS test is also illustrated by the analysis of thirty real data sets. Full article

The multimodal fusion of hyperspectral images (HSI) and LiDAR data for land cover classification encounters difficulties in modeling heterogeneous data characteristics and cross-modal dependencies, leading to the loss of complementary information due to concatenation, the inadequacy of fixed fusion weights to adapt to spatially varying reliability, and the assumptions of linear separability for nonlinearly coupled patterns. We propose QIE-Mamba, integrating selective state-space models with quantum-inspired processing to enhance multimodal representation learning. The framework employs ConvNeXt encoders for hierarchical feature extraction, quantum superposition layers for complex-valued multimodal encoding with learned amplitude–phase relationships, unitary entanglement networks via skew-symmetric matrix parameterization (validated through Cayley transform and matrix exponential methods), quantum-enhanced Mamba blocks with adaptive decoherence, and confidence-weighted measurement for classification. Systematic three-phase sequential validation on Houston2013, Muufl, and Augsburg datasets achieves overall accuracies of 99.62%, 96.31%, and 96.30%. Theoretical validation confirms 35.87% mutual information improvement over classical fusion (6.9966 vs. 5.1493 bits), with ablation studies demonstrating quantum superposition contributes 82% of total performance gains. Phase information accounts for 99.6% of quantum state entropy, while gradient convergence analysis confirms training stability (zero mean/std gradient norms). The optimization framework reduces hyperparameter search complexity by 99.6% while maintaining state-of-the-art performance. These results establish quantum-inspired state-space models as effective architectures for multimodal remote sensing fusion, providing reproducible methodology for hyperspectral–LiDAR classification with linear computational complexity. Full article

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

This paper develops a multiplicative model, termed the multiplicative Burr III distribution, by analyzing a parallel system consisting of two independently operating components, each having a Burr III-distributed lifetime. The multiplicative Burr III distribution is appropriate for modeling positively, negatively skewed, leptokurtic, platykurtic and over- and under-variation real data. A graphical illustration of the proposed model’s probability density, hazard rate, and reversed hazard rate functions is presented. The plots of the pdf and hrf of the multiplicative Burr III distribution exhibit approximately symmetric or unimodal shapes depending on the parameter values. This flexibility highlights the model’s capability to represent both symmetric and asymmetric behaviors in lifetime data. The fundamental characteristics of the multiplicative Burr III distribution are thoroughly established. Parameter estimation, along with the reliability, hazard rate, and reversed hazard rate functions, is conducted using the maximum likelihood approach. In addition, asymptotic confidence intervals are derived for the parameters and associated reliability and hazard functions. A comprehensive simulation study is performed to assess the efficiency of the maximum likelihood estimators. Finally, the practical relevance of the proposed distribution is validated through real-life datasets. Full article

This paper introduces the Marshall–Olkin unit-exponentiated-half-logistic (MO-UEHL) distribution, a novel three-parameter model designed to enhance the flexibility of the unit-exponentiated-half-logistic distribution through the incorporation of the Marshall–Olkin transformation. Defined on the unit interval $(0,1)$, the MO-UEHL distribution is well-suited for modeling proportional data exhibiting asymmetry. The Marshall–Olkin tilt parameter $\alpha$ explicitly controls the degree and direction of asymmetry, enabling the density to range from highly right-skewed to nearly symmetric unimodal forms, and even to left-skewed configurations for certain parameter values, thereby offering a direct mathematical representation of symmetry breaking in bounded proportional data. The resulting model achieves this versatility without relying on exponential terms or special functions, thus simplifying computational procedures. We derive its key mathematical properties, including the probability density function, cumulative distribution function, survival function, hazard rate function, quantile function, moments, and information-theoretic measures such as the Shannon and residual entropy. Parameter estimation is explored using maximum likelihood, maximum product spacing, ordinary and weighted least-squares, and Cramér–von Mises methods, with simulation studies evaluating their performance across varying sample sizes and parameter sets. The practical utility of the MO-UEHL distribution is demonstrated through applications to four real datasets from environmental and engineering contexts. The results highlight the MO-UEHL distribution’s potential as a valuable tool in reliability analysis, environmental modeling, and related fields. Full article

A nonparametric control chart is a type of control chart that does not rely on assumptions regarding the underlying distribution of the data. This characteristic provides greater flexibility and robustness, particularly when handling non-normal data, skewed distributions, or datasets containing outliers. The primary objective of this study is to propose a nonparametric TEWMA–MA control chart based on the sign statistic, designed to operate under both symmetric and asymmetric distributions for effective process monitoring. This chart aims to enhance the ability to quickly detect shifts in the production process. The run-length characteristics obtained through Monte Carlo simulation (MC) were employed as performance measures. In addition, overall efficiency was assessed using AEQL, RMI, and PCI. The proposed control chart was compared against MA, TEWMA, MA–TEWMA, TEWMA–MA, and MA–TEWMA sign charts. The findings indicate that the proposed chart is effective for process control and demonstrates superior detection capability compared to competing charts, particularly in identifying small to moderate shifts. Furthermore, to validate its practical utility, the proposed control chart was applied to real-world data. Full article

Consider the dynamical system x ˙ = f ( x ) , where x ∈ R n is the state vector, x ˙ is the time or spatial derivative, and f is the system model. We wish to identify unknown f from its time-series or spatial data. For this, we propose a Bayesian framework based on the maximum a posteriori (MAP) point estimate, to give a generalized Tikhonov regularization method with the residual and regularization terms identified, respectively, with the negative logarithms of the likelihood and prior distributions. As well as estimates of the model coefficients, the Bayesian interpretation provides access to the full Bayesian apparatus, including the ranking of models, the quantification of model uncertainties, and the estimation of unknown (nuisance) hyperparameters. For multivariate Gaussian likelihood and prior distributions, the Bayesian formulation gives a Gaussian posterior distribution, in which the numerator contains a Mahalanobis distance or “Gaussian norm”. In this study, two Bayesian algorithms for the estimation of hyperparameters—the joint maximum a posteriori (JMAP) and variational Bayesian approximation (VBA)—are compared to the popular SINDy, LASSO, and ridge regression algorithms for the analysis of several dynamical systems with additive noise. We consider two dynamical systems, the Lorenz convection system and the Shil’nikov cubic system, with four choices of noise model: symmetric Gaussian or Laplace noise and skewed Rayleigh or Erlang noise, with different magnitudes. The posterior Gaussian norm is found to provide a robust metric for quantitative model selection—with quantification of the model uncertainties—across all dynamical systems and noise models examined. Full article

Clustered time-to-event data are quite common in survival analysis and finding a suitable model to account for dispersion as well as censoring is an important issue. In this article, we present a flexible model for repeated, overdispersed time-to-event data with right-censoring. We present here a general model by incorporating generalized gamma and normal random effects in a Weibull distribution to accommodate overdispersion and data hierarchies, respectively. The normal random effect has the property of being symmetrical, which means its probability density function is symmetric around its mean. While the random effects are symmetrically distributed, the resulting frailty model is asymmetric in its survival function because the random effects enter the model multiplicatively via the hazard function, and the exponentiation of a symmetric normal variable leads to lognormal distribution, which is right-skewed. Due to the intractable integrals involved in the likelihood function and its derivatives, the Monte Carlo approach is used to approximate the involved integrals. The maximum likelihood estimates of the parameters in the model are then numerically determined. An extensive simulation study is then conducted to evaluate the performance of the proposed model and the method of inference developed here. Finally, the usefulness of the model is demonstrated by analyzing a data on recurrent asthma attacks in children and a recurrent bladder data set known in the survival analysis literature. Full article

Portfolio optimization is a cornerstone of modern financial decision-making, traditionally based on the mean–variance model introduced by Markowitz. However, this framework relies on restrictive assumptions—such as normally distributed returns and symmetric risk preferences—that often fail in real-world markets, particularly in volatile and non-Gaussian environments such as cryptocurrencies. To address these limitations, this paper proposes a novel multi-objective model that combines expected return maximization, mean absolute deviation (MAD) minimization, and entropy-based diversification into a unified optimization structure: the Mean–Deviation–Entropy (MDE) model. The MAD metric offers a robust alternative to variance by capturing the average magnitude of deviations from the mean without inflating extreme values, while entropy serves as an information-theoretic proxy for portfolio diversification and uncertainty. Three entropy formulations are considered—Shannon entropy, Tsallis entropy, and cumulative residual Sharma–Taneja–Mittal entropy (CR-STME)—to explore different notions of uncertainty and structural diversity. The MDE model is formulated as a tri-objective optimization problem and solved via scalarization techniques, enabling flexible trade-offs between return, deviation, and entropy. The framework is empirically tested on a cryptocurrency portfolio composed of Bitcoin (BTC), Ethereum (ETH), Solana (SOL), and Binance Coin (BNB), using daily data over a 12-month period. The empirical setting reflects a high-volatility, high-skewness regime, ideal for testing entropy-driven diversification. Comparative outcomes reveal that entropy-integrated models yield more robust weightings, particularly when tail risk and regime shifts are present. Comparative results against classical mean–variance and mean–MAD models indicate that the MDE model achieves improved diversification, enhanced allocation stability, and greater resilience to volatility clustering and tail risk. This study contributes to the literature on robust portfolio optimization by integrating entropy as a formal objective within a scalarized multi-criteria framework. The proposed approach offers promising applications in sustainable investing, algorithmic asset allocation, and decentralized finance, especially under high-uncertainty market conditions. Full article

This study explores advanced methods for analyzing the two-parameter alpha-power exponential (APE) distribution using data from a novel generalized progressive hybrid censoring scheme. The APE model is inherently asymmetric, exhibiting positive skewness across all valid parameter values due to its right-skewed exponential base, with the alpha-power transformation amplifying or dampening this skewness depending on the power parameter. The proposed censoring design offers new insights into modeling lifetime data that exhibit non-monotonic hazard behaviors. It enhances testing efficiency by simultaneously imposing fixed-time constraints and ensuring a minimum number of failures, thereby improving inference quality over traditional censoring methods. We derive maximum likelihood and Bayesian estimates for the APE distribution parameters and key reliability measures, such as the reliability and hazard rate functions. Bayesian analysis is performed using independent gamma priors under a symmetric squared error loss, implemented via the Metropolis–Hastings algorithm. Interval estimation is addressed using two normality-based asymptotic confidence intervals and two credible intervals obtained through a simulated Markov Chain Monte Carlo procedure. Monte Carlo simulations across various censoring scenarios demonstrate the stable and superior precision of the proposed methods. Optimal censoring patterns are identified based on the observed Fisher information and its inverse. Two real-world case studies—breast cancer remission times and global oil reserve data—illustrate the practical utility of the APE model within the proposed censoring framework. These applications underscore the model’s capability to effectively analyze diverse reliability phenomena, bridging theoretical innovation with empirical relevance in lifetime data analysis. Full article

The Maxwell–Boltzmann (MB) distribution is fundamental in statistical physics, providing an exact description of particle speed or energy distributions. In this study, a discrete formulation derived via the survival function discretization technique extends the MB model’s theoretical strengths to realistically handle lifetime and reliability data recorded in integer form, enabling accurate modeling under inherently discrete or censored observation schemes. The proposed discrete MB (DMB) model preserves the continuous MB’s flexibility in capturing diverse hazard rate shapes, while directly addressing the discrete and often censored nature of real-world lifetime and reliability data. Its formulation accommodates right-skewed, left-skewed, and symmetric probability mass functions with an inherently increasing hazard rate, enabling robust modeling of negatively skewed and monotonic-failure processes where competing discrete models underperform. We establish a comprehensive suite of distributional properties, including closed-form expressions for the probability mass, cumulative distribution, hazard functions, quantiles, raw moments, dispersion indices, and order statistics. For parameter estimation under Type-II censoring, we develop maximum likelihood, Bayesian, and bootstrap-based approaches and propose six distinct interval estimation methods encompassing frequentist, resampling, and Bayesian paradigms. Extensive Monte Carlo simulations systematically compare estimator performance across varying sample sizes, censoring levels, and prior structures, revealing the superiority of Bayesian–MCMC estimators with highest posterior density intervals in small- to moderate-sample regimes. Two genuine datasets—spanning engineering reliability and clinical survival contexts—demonstrate the DMB model’s superior goodness-of-fit and predictive accuracy over eleven competing discrete lifetime models. Full article

The entropy function, as a measure of information and uncertainty, has been widely applied in various scientific disciplines. One notable extension of entropy is exponential entropy, which finds applications in fields such as optimization, image segmentation, and fuzzy set theory. In this paper, we explore the continuous case of generalized exponential entropy and analyze its behavior under symmetric and asymmetric probability distributions. Particular emphasis is placed on illustrating the role of symmetry through analytical results and graphical representations, including comparisons of entropy curves for symmetric and skewed distributions. Moreover, we investigate the relationship between the proposed entropy model and other information-theoretic measures such as entropy and extropy. Several non-parametric estimation techniques are studied, and their performance is evaluated using Monte Carlo simulations, highlighting asymptotic properties and the emergence of normality, an aspect closely related to distributional symmetry. Furthermore, the consistency and biases of the estimation methods, which rely on kernel estimation with ${\rho }_{corr}$-mixing dependent data, are presented. Additionally, numerical calculations based on simulation and medical real data are applied. Finally, a test of uniformity using different test statistics is given. Full article

(1) Objective: Volatile organic compounds (VOCs) monitoring in industrial parks is crucial for environmental regulation and public health protection. However, current techniques face challenges related to cost and real-time performance. This study aims to develop a dynamic calibration framework for accurate real-time conversion of VOCs volume fractions (nmol mol⁻¹) to mass concentrations (μg m⁻³) in industrial environments, addressing the limitations of conventional monitoring methods such as high costs and delayed response times. (2) Methods: By innovatively integrating photoionization detector (PID) with machine learning, we developed a robust conversion model utilizing PID signals, meteorological data, and a random forest’s (RF) algorithm. The system’s performance was rigorously evaluated against standard gas chromatography-flame ionization detectors (GC-FID) measurements. (3) Results: The proposed framework demonstrated superior performance, achieving a coefficient of determination (R²) of 0.81, root mean squared error (RMSE) of 48.23 μg m⁻³, symmetric mean absolute percentage error (SMAPE) of 62.47%, and a normalized RMSE (RMSE_norm) of 2.07%, outperforming conventional methods. This framework not only achieved minute-level response times but also reduced costs to just 10% of those associated with GC-FID methods. Additionally, the model exhibited strong cross-site robustness with R² values ranging from 0.68 to 0.69, although its accuracy was somewhat reduced for high-concentration samples (>1500 μg m⁻³), where the mean absolute percentage error (MAPE) was 17.8%. The inclusion of SMAPE and RMSE_norm provides a more nuanced understanding of the model’s performance, particularly in the context of skewed or heteroscedastic data distributions, thereby offering a more comprehensive assessment of the framework’s effectiveness. (4) Conclusions: The framework’s innovative combination of PID’s real-time capability and RF’s nonlinear modeling achieves accurate mass concentration conversion (R² = 0.81) while maintaining a 95% faster response and 90% cost reduction compared to GC-FID systems. Compared with traditional single-coefficient PID calibration, this framework significantly improves accuracy and adaptability under dynamic industrial conditions. Future work will apply transfer learning to improve high-concentration detection for pollution tracing and environmental governance in industrial parks. Full article

Many scholars are interested in modeling complex data in an effort to create novel probability distributions. This article proposes a novel class of distributions based on the inverse of the exponentiated Weibull hazard rate function. A particular member of this class, the Weibull–Rayleigh distribution (WR), is presented with focus. The WR features diverse probability density functions, including symmetric, right-skewed, left-skewed, and the inverse J-shaped distribution which is flexible in modeling lifetime and systems data. Several significant statistical features of the suggested WR are examined, covering the quantile, moments, characteristic function, probability weighted moment, order statistics, and entropy measures. The model accuracy was verified through Monte Carlo simulations of five different statistical estimation methods. The significance of WR is demonstrated with three real-world data sets, revealing a higher goodness of fit compared to other competing models. Additionally, the change point for the WR model is illustrated using the modified information criterion (MIC) to identify changes in the structures of these data. The MIC and curve analysis captured a potential change point, supporting and proving the effectiveness of WR distribution in describing transitions. Full article

This paper introduces a novel and flexible class of continuous probability distributions, termed the Alpha Power Survival Ratio-X (APSR-X) family. Unlike many existing transformation-based families, the APSR-X class integrates an alpha power transformation with a survival ratio structure, offering a new mechanism for enhancing shape flexibility while maintaining mathematical tractability. This construction enables fine control over both the tail behavior and the symmetry properties, distinguishing it from traditional alpha power or survival-based extensions. We focus on a key member of this family, the two-parameter Alpha Power Survival Ratio Exponential (APSR-Exp) distribution, deriving essential mathematical properties including moments, quantile functions and hazard rate structures. We estimate the model parameters using eight frequentist methods: the maximum likelihood (MLE), maximum product of spacings (MPSE), least squares (LSE), weighted least squares (WLSE), Anderson–Darling (ADE), right-tailed Anderson–Darling (RADE), Cramér–von Mises (CVME) and percentile (PCE) estimation. Through comprehensive Monte Carlo simulations, we evaluate the estimator performance using bias, mean squared error and mean relative error metrics. The proposed APSR-X framework uniquely enables preservation or controlled modification of the symmetry in probability density and hazard rate functions via its shape parameter. This capability is particularly valuable in reliability and survival analyses, where symmetric patterns represent balanced risk profiles while asymmetric shapes capture skewed failure behaviors. We demonstrate the practical utility of the APSR-Exp model through three real-world applications: economic (tax revenue durations), engineering (mechanical repair times) and medical (infection durations) datasets. In all cases, the proposed model achieves a superior fit over that of the conventional alternatives, supported by goodness-of-fit statistics and visual diagnostics. These findings establish the APSR-X family as a unique, symmetry-aware modeling framework for complex lifetime data. Full article