Search Results (46)

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

This study introduces a new and flexible class of probability distributions known as the novel alpha-power X (NAP-X) family. A key development within this framework is the novel alpha-power half-logistic (NAP-HL) distribution, which extends the classical half-logistic model through an alpha-power transformation, allowing for greater adaptability to various data shapes. The paper explores several theoretical aspects of the proposed model, including its moments, quantile function and hazard rate. To assess the effectiveness of parameter estimation, a detailed simulation study is conducted using seven estimation techniques: Maximum likelihood estimation (MLE), Cramér–von Mises estimation (CVME), maximum product of spacings estimation (MPSE), least squares estimation (LSE), weighted least squares estimation (WLSE), Anderson–Darling estimation (ADE) and a right-tailed version of Anderson–Darling estimation (RTADE). The results offer comparative insights into the performance of each method across different sample sizes. The practical value of the NAP-HL distribution is demonstrated using two real datasets from the metrology and engineering domains. In both cases, the proposed model provides a better fit than the traditional half-logistic and related distributions, as shown by lower values of standard model selection criteria. Graphical tools such as fitted density curves, Q–Q and P–P plots, survival functions and box plots further support the suitability of the model for real-world data analysis. Full article

Probability models are instrumental in a wide range of applications by being able to accurately model real-world data. Over time, numerous probability models have been developed and applied in practical scenarios. This study introduces the AAP-X family of distributions—a novel, flexible framework for continuous data analysis named after authors Aadil Ajaz and Parvaiz. The proposed family effectively accommodates both symmetric and asymmetric characteristics through its shape-controlling parameter, an essential feature for capturing diverse data patterns. A specific subclass of this family, termed the “AAP Exponential” (AAPEx) model is designed to address the inflexibility of classical exponential distributions by accommodating versatile hazard rate patterns, including increasing, decreasing and bathtub-shaped patterns. Several fundamental mathematical characteristics of the introduced family are derived. The model parameters are estimated using six frequentist estimation approaches, including maximum likelihood, Cramer–von Mises, maximum product of spacing, ordinary least squares, weighted least squares and Anderson–Darling estimation. Monte Carlo simulations demonstrate the finite-sample performance of these estimators, revealing that maximum likelihood estimation and maximum product of spacing estimation exhibit superior accuracy, with bias and mean squared error decreasing systematically as the sample sizes increases. The practical utility and symmetric–asymmetric adaptability of the AAPEx model are validated through five real-world applications, with special emphasis on cancer survival times, COVID-19 mortality rates and reliability data. The findings indicate that the AAPEx model outperforms established competitors based on goodness-of-fit metrics such as the Akaike Information Criteria (AIC), Schwartz Information Criteria (SIC), Akaike Information Criteria Corrected (AICC), Hannan–Quinn Information Criteria (HQIC), Anderson–Darling (A*) test statistic, Cramer–von Mises (W*) test statistic and the Kolmogorov–Smirnov (KS) test statistic and its associated p-value. These results highlight the relevance of symmetry in real-life data modeling and establish the AAPEx family as a powerful tool for analyzing complex data structures in public health, engineering and epidemiology. Full article

This paper investigates various estimation methods for the parameters of the unit Lindley distribution (U-LD) under both ranked set sampling (RSS) and simple random sampling (SRS) designs. The distribution parameters are estimated using the maximum likelihood estimation, ordinary least squares, weighted least squares, maximum product of spacings, minimum spacing absolute distance, minimum spacing absolute log-distance, minimum spacing square distance, minimum spacing square log-distance, linear-exponential, Anderson–Darling (AD), right-tail AD, left-tail AD, left-tail second-order, Cramér–von Mises, and Kolmogorov–Smirnov. A comprehensive simulation study is conducted to assess the performance of these estimators, ensuring an equal number of measuring units across both designs. Additionally, two real datasets of items failure time and COVID-19 are analyzed to illustrate the practical applicability of the proposed estimation methods. The findings reveal that RSS-based estimators consistently outperform their SRS counterparts in terms of mean squared error, bias, and efficiency across all estimation techniques considered. These results highlight the advantages of using RSS in parameter estimation for the U-LD distribution, making it a preferable choice for improved statistical inference. 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

Given the increasing number of phenomena that demand interpretation and investigation, developing new distributions and families of distributions has become increasingly essential. This article introduces a novel family of distributions based on the exponentiated reciprocal of the hazard rate function named the new Lomax-G family of distributions. We demonstrate the family’s flexibility to predict a wide range of lifetime events by deriving its cumulative and probability density functions. The new Lomax–Weibull distribution (NLW) is studied as a sub-model, with analytical and graphical evidence indicating its efficiency for reliability analysis and complex data modeling. The NLW density encompasses a variety of shapes, such as symmetrical, semi-symmetrical, right-skewed, left-skewed, and inverted J shapes. Furthermore, its hazard function exhibits a broad range of asymmetric forms. Five estimation techniques for determining the parameters of the proposed NLW distribution include the maximum likelihood, percentile, least squares, weighted least squares, and Cramér–von Mises methods. The performance of the estimators of the studied inferential methods is investigated through a comparative Monte Carlo simulation study and numerical demonstration. Additionally, the effectiveness of the NLW is validated by means of four real-world datasets. The results indicate that the NLW distribution provides a more accurate fit than several competing models. Full article

The Cramér–von Mises test provides a useful criterion for assessing goodness of fit in various problems. In this paper, we introduce a novel Cramér–von Mises-type test for testing distributions of high-dimensional continuous data. We establish an asymptotic theory for the proposed test statistics based on quadratic functions in high-dimensional stochastic processes. To estimate the limiting distribution of the test statistic, we propose two practical approaches: a plug-in calibration method and a subsampling method. Theoretical justifications are provided for both techniques. Numerical simulation also confirms the convergence of the proposed methods. Full article

This article defines a new distribution using a novel alpha power-transformed method extension. The model obtained has three parameters and is quite effective in modeling skewed, complex, symmetric, and asymmetric datasets. The new approach has one additional parameter for the model. Certain distributional and mathematical properties are investigated, notably reliability, quartile, moments, skewness, kurtosis, and order statistics, and several approaches of estimation, notably the maximum likelihood, least square, weighted least square, maximum product spacing, Cramer-Von Mises, and Anderson Darling estimators of the model parameters were obtained. A Monte Carlo simulation study was conducted to evaluate the performance of the proposed techniques of estimation of the model parameters. The actuarial measures are computed for our recommended model. At the end of the paper, two insurance applications are illustrated to check the potential and utility of the suggested distribution. Evaluation using four selection criteria indicates that our recommended model is the most appropriate probability model for modeling insurance datasets. Full article

Different non-Bayesian and Bayesian techniques were used to estimate the pseudo-Lindley (PsL) distribution’s parameters in this study. To derive Bayesian estimators, one must assume appropriate priors on the parameters and use loss functions such as squared error (SE), general entropy (GE), and linear-exponential (LINEX). Since no closed-form solutions are accessible for Bayes estimates under these loss functions, the Markov Chain Monte Carlo (MCMC) approach was used. Simulation studies were conducted to evaluate the estimators’ performance under the given loss functions. Furthermore, we exhibited the adaptability and practicality of the PsL distribution through real-world data applications, which is essential for evaluating the various estimation techniques. Also, the acceptance sampling plans were developed in this work for items whose lifespans approximate the PsL distribution. Full article

In this paper, we introduce a new three-parameter inverse Rayleigh distribution that extends the inverse Rayleigh distribution, constructed based on the generalized transmuted family of distributions proposed by Alizadeh, Merovci, and Hamedani. We explore statistical properties such as the quantile function, moments, harmonic mean, mean deviation, stress–strength reliability, and entropy. Parameter estimation is performed using various methods, including maximum likelihood, least squares, the method of the maximum product of spacings, and the method of Cramér–von Mises. The usefulness of the new three-parameter inverse Rayleigh distribution is illustrated by modeling a real dataset, demonstrating its superior fit compared to several other distributions. Full article

Modeling diameter distribution is a crucial aspect of forest management, requiring the selection of an appropriate probability density function or cumulative distribution function along with a fitting method. This study compared the suitability of eight probability density functions—A Charlier, beta, generalized beta, gamma, Gumbel, Johnson’s SB, and Weibull (two- and three-parameter)—fitted using both derivative methods (Moments) fitted in SAS/STAT^TM and optimization methods (MLE) fitted with the ‘optim’ function in R for diameter distribution estimation in forest stands. The A Charlier and Gumbel functions were used for the first time in this type of comparison. The data were derived from 167 permanent sample plots in an Atlantic forest (Quercus robur) and 59 temporary sample plots in tropical forests (Tectona grandis). Fit quality was assessed using various indices, including Kolmogorov–Smirnov, Cramér–von Mises, mean absolute error, bias, and mean squared error. The results indicated that Johnson’s SB function was more suitable for describing the diameter distribution of the stands. Johnson’s SB, three-parameter Weibull, and generalized beta consistently performed well across different fitting methods, while the fits produced by gamma, Gumbel, and two-parameter Weibull were of poor quality. Full article

In this study, we introduce a new generalized family of distributions called the Exponentiated Half Logistic-Harris-G (EHL-Harris-G) distribution, which extends the Harris-G distribution. The motivation for introducing this generalized family of distributions lies in its ability to overcome the limitations of previous families, enhance flexibility, improve tail behavior, provide better statistical properties and find applications in several fields. Several statistical properties, including hazard rate function, quantile function, moments, moments of residual life, distribution of the order statistics and Rényi entropy are discussed. Risk measures, such as value at risk, tail value at risk, tail variance and tail variance premium, are also derived and studied. To estimate the parameters of the EHL-Harris-G family of distributions, the following six different estimation approaches are used: maximum likelihood (MLE), least-squares (LS), weighted least-squares (WLS), maximum product spacing (MPS), Cramér–von Mises (CVM), and Anderson–Darling (AD). The Monte Carlo simulation results for EHL-Harris-Weibull (EHL-Harris-W) show that the MLE method allows us to obtain better estimates, followed by WLS and then AD. Finally, we show that the EHL-Harris-W distribution is superior to some other equi-parameter non-nested models in the literature, by fitting it to two real-life data sets from different disciplines. Full article

This paper introduces a flexible three-parameter extension of the Lomax model called the odd Lomax–Lomax (OLxLx) distribution. The OLxLx distribution can provide left-skewed, symmetrical, right-skewed, and reversed-J shaped densities and increasing, constant, unimodal, and decreasing hazard rate shapes. Some mathematical properties of the introduced model are derived. The OLxLx density can be expressed as mixture of Lomax densities. The OLxLx parameters are estimated by using eight estimation methods and their performance is explored by using detailed simulation studies. The partial and overall ranks of the mean relative errors, absolute biases, and mean square errors of different estimators are presented to choose the best estimation method. The flexibility and applicability of the OLxLx distribution is shown using real-life medicine data, illustrating the superior fit of the OLxLx distribution over other competing Lomax distributions. The OLxLX distribution outperforms some rival Lomax distributions including the Kumaraswamy–Lomax, McDonald–Lomax, Weibull–Lomax, transmuted Weibull–Lomax, exponentiated-Lomax, Lomax–Weibull, modified Kies–Lomax, Burr X Lomax, beta exponentiated-Lomax, odd exponentiated half-logistic Lomax, and transmuted-Lomax distributions, among others. Full article

In this paper, we emphasize a new one-parameter distribution with support as $[1,+\infty )$. It is constructed from the inverse method applied to an understudied one-parameter unit distribution, the unit Teissier distribution. Some properties are investigated, such as the mode, quantiles, stochastic dominance, heavy-tailed nature, moments, etc. Among the strengths of the distribution are the following: (i) the closed-form expressions and flexibility of the main functions, and in particular, the probability density function is unimodal and the hazard rate function is increasing or unimodal; (ii) the manageability of the moments; and, more importantly, (iii) it provides a real alternative to the famous Pareto distribution, also with support as $[1,+\infty )$. Indeed, the proposed distribution has different functionalities but also benefits from the heavy-right-tailed nature, which is demanded in many applied fields (finance, the actuarial field, quality control, medicine, etc.). Furthermore, it can be used quite efficiently in a statistical setting. To support this claim, the maximum likelihood, Anderson–Darling, right-tailed Anderson–Darling, left-tailed Anderson–Darling, Cramér–Von Mises, least squares, weighted least-squares, maximum product of spacing, minimum spacing absolute distance, and minimum spacing absolute-log distance estimation methods are examined to estimate the unknown unique parameter. A Monte Carlo simulation is used to compare the performance of the obtained estimates. Additionally, the Bayesian estimation method using an informative gamma prior distribution under the squared error loss function is discussed. Data on the COVID mortality rate and the timing of pain relief after receiving an analgesic are considered to illustrate the applicability of the proposed distribution. Favorable results are highlighted, supporting the importance of the findings. Full article

In this paper, the estimation of the stress–strength reliability is taken into account when the stress and strength variables have unit Gompertz distributions with a similar scale parameter. The consideration of the unit Gompertz distribution in this context is because of its intriguing symmetric and asymmetric properties that can accommodate various histogram proportional-type data shapes. As the main contribution, the reliability estimate is determined via seven frequentist techniques using the ranked set sampling (RSS) and simple random sampling (SRS). The proposed methods are the maximum likelihood, least squares, weighted least squares, maximum product spacing, Cramér–von Mises, Anderson–Darling, and right tail Anderson–Darling methods. We perform a simulation work to evaluate the effectiveness of the recommended RSS-based estimates by using accuracy metrics. We draw the conclusion that the reliability estimates in the maximum product spacing approach have the lowest value compared to other approaches. In addition, we note that the RSS-based estimates are superior to those obtained by a comparable SRS approach. Additional results are obtained using two genuine data sets that reflect the survival periods of head and neck cancer patients. Full article