Search Results (56)

Selecting an appropriate statistical model to describe lifetime data with heterogeneous characteristics poses a considerable challenge due to the diverse distributional features exhibited by real-world datasets. One effective way to address this challenge is to develop flexible families of distributions capable of generating numerous member distributions. In this paper, the Transformed–Transformer framework is employed to introduce a new and more flexible class of continuous distributions, referred to as the Canonical Type II Unitary Weibull–H family. To illustrate the flexibility of the proposed family, four representative sub-models are derived. Among them, the Canonical Type II Unitary Weibull–Chen distribution is selected for a comprehensive theoretical and empirical investigation. The general properties of the proposed Canonical Type II Unitary Weibull–H family are established, while a detailed study of the Canonical Type II Unitary Weibull–Chen distribution is provided, including mixture representations, the quantile function, moments, probability-weighted moments, Rényi entropy, and order statistics. Moreover, five classical estimation methods, namely maximum likelihood, least squares, weighted least squares, maximum product spacing, and Cramér–von Mises estimation, are employed to estimate the model parameters. In addition, an extensive Monte Carlo simulation study is conducted to evaluate and compare the finite-sample performance of the different estimators. Finally, the practical usefulness of the proposed distribution is illustrated through the analysis of four real-world datasets representing waiting time, biomedical, survival, and reliability applications. Comparisons based on several goodness-of-fit criteria demonstrate that the Canonical Type II Unitary Weibull–Chen distribution provides a competitive and, in many cases, superior fit relative to several well-established competing models. Full article

Lifetime data arising in engineering reliability, survival analysis, actuarial science, and environmental studies often exhibit substantial right-skewness, extreme observations, and heterogeneous hazard-rate structures. Classical lifetime distributions may not adequately capture these characteristics, thereby affecting risk assessment, reliability evaluation, and predictive performance. In this paper, we introduce the Heavy-Tailed QLindley (HTQL) distribution, a new two-parameter heavy-tailed extension of the QLindley model obtained through the New Family of Heavy-Tailed (NFHT) transformation. The proposed distribution provides greater flexibility for modelling positively skewed and heavy-tailed data while preserving analytical tractability. The HTQL model accommodates increasing, decreasing, bathtub-shaped, unimodal, and nearly constant hazard rate functions, making it suitable for applications in reliability analysis, survival studies, actuarial science, and environmental modelling. Several mathematical and statistical properties of the HTQL distribution are derived, including explicit expressions for the quantile function, ordinary and incomplete moments, order statistics, and reliability measures. Important tail-based risk measures such as Value-at-Risk, Tail Value-at-Risk, Tail Variance, Tail Variance Premium, and Expected Shortfall are also obtained. Parameter estimation is investigated using maximum likelihood, ordinary least squares, Cramér–von Mises, and Bayesian approaches, together with bootstrap confidence intervals. A Monte Carlo simulation study is conducted to evaluate the finite-sample performance of the proposed estimators. The practical usefulness of the HTQL distribution is illustrated using three real-world datasets from pharmacokinetics, engineering reliability, and environmental studies. The empirical results show that the HTQL distribution provides highly competitive fits compared with several classical, Lindley-type, and heavy-tailed distributions. Overall, the proposed model constitutes a flexible and parsimonious alternative for modelling positive heavy-tailed data. Full article

The Poisson–Three-Parameter Lindley (PTPL) distribution constitutes a flexible Poisson mixture model for overdispersed count data, encompassing several classical count distributions as special or limiting cases. Despite its growing use in applied contexts, no formal goodness-of-fit test specifically designed for this distribution is currently available. In this paper, we propose and study a new goodness-of-fit test for the PTPL model based on a Cramér–von Mises type distance between the empirical and theoretical probability generating functions (PGFs). For polynomial weight functions, the test statistic admits an explicit closed-form representation; in practice, it is computed efficiently via numerical quadrature. The null distribution of the statistic is approximated via parametric bootstrap. We establish theoretical properties of the proposed procedure, including consistency against fixed alternatives and the validity of the bootstrap approximation. Monte Carlo simulations with sample sizes $n\in \{50, 100, 150, 200, 500\}$ for size evaluation and $n\in \{100, 250, 500\}$ for power comparisons, as well as weight exponents $a\in \{0, 1, 2\}$, show that the empirical size is well controlled at both the 5% and 10% nominal levels, and that the test exhibits competitive power against Poisson, Negative Binomial, COM-Poisson, and Zero-Inflated Poisson alternatives. A real data application to five overdispersed count datasets further illustrates the practical utility of the method. The empirical size is further verified across twelve parameter configurations spanning dispersion indices from 1.37 to 59.33, confirming bootstrap validity under strong overdispersion. Full article

This study presents a novel flexible Rayleigh–exponential mixture detection model (REMDM) for estimating population abundance under line transect sampling. The proposed detection function combines a Rayleigh-type component with an exponential component to provide greater flexibility in modelling perpendicular distance data and capturing the complex detection patterns commonly observed in ecological surveys. The model exhibited smooth behaviour near the transect line and flexible tail decay, making it suitable for heterogeneous detection structures. Several statistical properties of the proposed REMDM were derived, including the probability density function, cumulative distribution function, moments, and hazard rate function. Parameters were estimated by using the maximum likelihood estimation method. The performance of the estimators is evaluated through extensive Monte Carlo simulation studies under various sample sizes and parameter settings. The simulation results indicate that the proposed estimators are consistent and efficient in terms of bias and mean squared error, with improved performance as the sample size increases. The applicability of the proposed model is demonstrated using a real perpendicular distance dataset and model performance is assessed using several goodness-of-fit measures, including the Akaike Information Criterion, Bayesian Information Criterion, Kolmogorov–Smirnov statistic, Anderson–Darling statistic, and Cramér–von Mises statistic. The results show that the REMDM provides a superior fit to several existing detection functions. In general, the proposed model offers a flexible and effective alternative for modelling detection probability and improving population abundance estimates in ecological distance sampling. Full article

The purpose of our paper is to provide a family of bilinear orthogonal expansions all based upon the same general pattern that is valid for a wide class of special functions. Our first family involves Jacobi, Laguerre, and Hermite polynomials. We give a discrete analogue of these bilinear expansions, the three families of classical orthogonal polynomials being replaced by zonal spherical functions associated with regular distance graphs. Such expansions playing a key role in the field of mathematical statistics, we show how our results apply to this field. We provide generalizations of the well-known Cramér–von Mises and Watson’s statistics, based upon an interpretation of their kernel in terms of the circular Laplacian. The product formula, well-known for zonal functions on Lie groups, is stated for distance-regular graphs, providing an elegant tool for proofs. Examples involving Hahn, q-Hahn, and Krawtchouk polynomials are given. Full article

This study investigates a range of parameter estimation methods for the Half-Logistic Inverse Rayleigh Distribution (HLIRD) under two distinct sampling frameworks: ranked set sampling (RSS) and simple random sampling (SRS). The estimation techniques considered include maximum likelihood estimation, ordinary and weighted least squares, and the maximum and minimum product of spacings methods. Model adequacy is evaluated using five goodness-of-fit criteria: the Anderson–Darling (AD) statistic, its right- and left-tail variants, the second-order left-tail AD statistic, and the Cramér–von Mises statistic. An extensive simulation study is conducted to thoroughly evaluate and compare the performance of the proposed estimators while maintaining a fixed total number of observations across both sampling schemes. The practical relevance of the proposed methods is further illustrated through an application to a real dataset consisting of 69 carbon fiber specimens, with tensile strength measurements (in GPa) recorded at a gauge length of 20 mm. The numerical results demonstrate that estimators based on RSS consistently outperform their SRS counterparts across all considered performance measures, including mean squared error, bias, and mean absolute relative error. Overall, the findings highlight the advantages of employing RSS for parameter estimation of the HLIRD, particularly due to its superior efficiency in small-sample scenarios. Full article

The fundamental problem with current Rayleigh-Lomax-based distributions lies in their limited flexibility to model both symmetry and tail weight simultaneously. Therefore, this study aims to introduce the OGRLx anomalous general distribution as an innovative mathematical framework that addresses these shortcomings by providing precise control over the distribution’s shape and risk ratios. We derived the basic statistical properties of the model, and used six different estimation methods that proved their efficiency through an intensive simulation study, with the Maximum Likelihood Estimator showing the best performance in terms of bias criteria and root mean square error. The practical value of the model is evident in its superior ability to fit data with high skewness and variable risks; experimental results using economic and medical data (bladder cancer) have proven the OGRLx distribution to be significantly superior to nine competing models. It achieved the lowest values for information standards Akaike Information Criteria, Consistent AIC, Bayesian Information Criteria, Hanan and Quinn Information Criteria, Anderson–Darling, Cramer–von Mises, Kolmogorov–Smirnov, and the highest p-value tests, making it a more accurate statistical tool for reliability analysis and medical studies compared to traditional extensions. Finally, it should be noted that all analyses, programming, and statistical operations in this study were performed using the R statistical software. Full article

A new generalized family of distributions, termed the Cosine–Topp–Leone–Exponentiated Half Logistic–G (Cos–TL–EHL–G) family, is proposed. The primary motivation for introducing this family is to enhance the modelling flexibility of the existing Cosine–Topp–Leone–G class by incorporating a exponentiated half logistic (EHL-G)-based transformation. Two important special cases, namely the Cos–TL–EHL–Weibull (Cos–TL–EHL–W) and Cos–TL–EHL–Log–Logistic (Cos–TL–EHL–LLoG) distributions, are presented. Several mathematical and statistical properties of the proposed family are derived, including series expansions, moments, order statistics, and uncertainty measures. Parameter estimation is carried out using maximum likelihood, least squares, Anderson–Darling, and Cramér–von Mises methods. A Monte Carlo simulation study indicates that the maximum likelihood estimator outperforms the competing estimation techniques. The practical usefulness and robustness of the proposed family are illustrated through applications to two real datasets, where the Cos–TL–EHL–W distribution demonstrates superior performance compared to both nested and non-nested competing models. Full article

This paper introduces the Topp–Leone Heavy-Tailed Odd Burr X-G (TL-HT-OBX-G) family of distributions (FOD), designed to model diverse data patterns. The new distribution is an infinite linear combination of the established exponentiated-G distributions. We used the established properties of the exponentiated-G distribution to infer the properties of the new FOD. The properties considered include the quantile function, moments and moment generating functions, probability-weighted moments, order statistics, stochastic orderings, and Rényi entropy. Parameter estimation is performed using multiple techniques, such as maximum likelihood, least squares, weighted least squares, Anderson–Darling, Cramér–von Mises, and Right-Tail Anderson–Darling. The maximum likelihood estimation method produced superior results in the Monte Carlo simulation studies. A special case of the developed model was applied to three real-world datasets. The model parameters were estimated using the maximum likelihood method. The selected special model was compared to other competing models, and goodness-of-fit was evaluated by the use of several goodness-of-fit statistics. The developed model fit the selected real-world datasets better than all the selected competing models. The new FOD provides a new framework for data modeling in health sciences and reliability datasets. Full article

Climate change increases the intensity and frequency of extreme rainfall. Heavy rain is one of the main input sources for the complex water resources system in the watershed. Understanding its regional spatial correlation is of vital importance for promoting sustainable disaster management in the watershed. The Qiantang River Basin is a significant ecological and economic area in the Yangtze River Delta, yet systematic research on its multi-regional rainstorm-dependent structure remains insufficient. In this study, hourly rainfall data of the basin from 1950 to 2024 were used to construct marginal functions by using the peaks-over-threshold and the generalized Pareto distribution, and a mixed Copula model was established to describe the dependence structure of multi-regional extreme rainfall events. The model has been tested by RMSE and Cramér–von Mises statistics and shows reliable performance. The study reveals that the basin has a “double cluster” spatial pattern: the internal conditions of northern clusters (Hangzhou–Shaoxing) and southern clusters (Jinhua–Lishui–Quzhou) showed a strong dependence. On the contrary, under cluster conditions with low inter-regional dependence, all high-probability combinations occurred within the clusters, not outside them. This finding provides quantitative support for optimizing trans-regional emergency response, improving flood control resilience, and realizing precise allocation of resources, and is of great significance for promoting sustainable watershed governance. Full article

Competing risks modeling plays a pivotal role in both reliability analysis for scientific and engineering fields and survival analysis within medical research. In real-world scenarios, failure or death (from a biological perspective) often arises from multiple risk factors that compete with one another. To adequately capture these complexities, it is essential to employ a flexible probabilistic framework, such as the competing risks model, which ensures suitability for intricate risk scenarios (e.g., analyzing data from aggressive diseases where treatment response and disease progression are closely interwoven). This study introduces Stacy’s competing risks model, built upon Stacy’s generalized gamma distribution, offering enhanced robustness and flexibility over existing models. The paper first develops the mathematical properties of the proposed model, followed by a detailed exploration of parameter estimation through various estimation methods. A key focus is accurately estimating shape parameters to gain deeper insights into the survival and failure mechanisms associated with the underlying phenomenon. The performance of different estimation approaches is assessed using Monte Carlo simulations, with results indicating that the least square, Cramér–von Mises, Anderson–Darling, right Anderson–Darling, and weighted least square had better performance and stable estimation accuracy compared with maximum likelihood maximum product of spacings methods. The model is applied to two real-world blood cancer datasets to demonstrate practical applicability, showing the superior performance and outstanding fit of the Anderson–Darling method among the other methods. The findings highlight the superior performance of Stacy’s competing risks model, supported by low Kolmogorov–Smirnov statistics and high p-values, affirming its suitability and robustness in modeling blood cancer data compared to other standard models. Full article

This paper presents a new model that surpasses traditional distributions, specifically the three-parameter distribution of the Inverse Power Entropy Chen (IPEC) model. In comparison to the existing distributions, the latest one presents an exceptionally diverse array of probability functions. The density and hazard rate functions have characteristics indicating that the model is adaptable to many types of data. The study explores the mathematical features of the IPEC distribution, including moments with some related measures, quantile function, Rényi entropy, Tsallis entropy, and order statistics. To estimate the parameters of the IPEC model, we utilized seven classical estimation strategies, including maximum likelihood estimators, Anderson–Darling estimators, right-tail Anderson–Darling estimators, Cramér–von Mises estimators, percentile estimators, least-squares estimators, and weighted least-squares estimators. To evaluate the efficacy of these estimating approaches across varying sample sizes, Monte Carlo simulations are performed. The efficacy of each estimator is evaluated through comparisons of average relative bias and mean squared error, highlighting their suitability for the used samples. Three applications utilize real-world datasets related to medical and physical fields, demonstrating the usefulness of the new model in relation to several established competitive models. This empirical investigation further supports the utility and adaptability of the inverse power entropy Chen model in capturing the intricacies of distinct datasets, hence delivering useful insights for practitioners in numerous domains. Full article

Along with other order statistics, the Cramér–von Mises (CM) statistic can assess the goodness of fit. CM does not have an explicit formula for the cumulative distribution function and the alternate way is to obtain its critical value from a Monte Carlo (MC) experiment. A high resolution experiment was deployed to generate a large amount of data resembling CM. Twenty-one repetitions of the experiment were conducted, and in each case, critical values of the CM statistic were obtained for all permilles and sample sizes from 2 to 30. The raw data presented here can serve to interpolate and extract probabilities associated with CM statistic directly, or to obtain a mathematical model for the bivariate dependence. 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

Increasing the amount of data with complex dynamics requires the constant updating of statistical distributions. This study aimed to introduce a new three-parameter distribution, named the new exponentiated Weibull (NEW) distribution, by applying the logarithmic transformation to the exponentiated Weibull distribution. The exponentiated Weibull distribution is a powerful generalization of the Weibull distribution that includes several classical distributions as special cases—Weibull, exponential, Rayleigh, and exponentiated exponential—which make it capable of capturing diverse forms of hazard functions. By combining the advantages of the logarithmic transformation and exponentiated Weibull, the new distribution offers great flexibility in modeling different forms of hazard functions, including increasing, J-shaped, reverse-J-shaped, and bathtub-shaped functions. Some mathematical properties of the NEW distribution were studied. Moreover, four different methods of estimation—the maximum likelihood (ML), least squares (LS), Cramer–Von Mises (CVM), and percentile (PE) methods—were employed to estimate the distribution parameters. To assess the performance of the estimates, three simulation studies were conducted, showing the benefit of the ML method, followed by the PE method, in estimating the model parameters. Additionally, five datasets were used to evaluate the effectiveness of the new distribution in fitting real data. Compared with some Weibull-type extensions, the results demonstrate the superiority of the new distribution in modeling various forms of real data and provide evidence for the applicability of the new distribution. Full article