Search Results (169)

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

The variance function (VF) is central to natural exponential family (NEF) theory. Prompted by an online query about whether, beyond the classical normal NEF, other real-line NEFs with bounded VFs exist, we establish three complementary sets of sufficient conditions that yield many such families. One set imposes a polynomial-growth bound on the NEF’s generating measure, ensuring rapid tail decay and a uniformly bounded VF. A second set relies on the Legendre duality, requiring a uniform positive lower bound on the second derivative of the generating function, which likewise ensures a bounded VF. The third set starts from the standard normal distribution and constructs an explicit sequence of NEFs whose Laplace transforms and VFs remain bounded. Collectively, these results reveal a remarkably broad class of NEFs whose Laplace transforms are not expressible in elementary form (apart from those stemming from the standard normal case), yet can be handled easily using modern symbolic and numerical software. Worked examples show that NEFs with bounded VFs are far more varied than previously recognized, offering practical alternatives to the normal and other classical models for real-data analysis across many fields. Full article

In this paper, we introduce a new distribution for modeling bimodal data supported on non-negative real numbers and particularly suited with an excess of very small values. This family of distributions is derived by multiplying the exponential distribution by a fourth-degree polynomial, resulting in a model that better fits the shape of the second mode of the empirical distribution of the data. We study the general density of this new family of distributions, along with its properties, moments, and skewness and kurtosis coefficients. A simulation study is performed to estimate parameters by the maximum likelihood method. Additionally, we present two applications to real-world datasets, demonstrating that the new distribution provides a better fit than the bimodal exponential distribution. Full article

This paper introduces a new extension of exponentiated standard logistic distribution. Some important statistical properties of the novel family of distributions are discussed. A simulation study is also conducted to observe the behavior of the estimated parameter using several estimation methods. The adaptability as well as the flexibility of the new model is checked through two real-life applications. A comprehensive financial risk assessment is conducted using multiple actuarial risk measures: Peaks Over Random Threshold Value-at-Risk, Value-at-Risk, Tail Value-at-Risk, the risk-adjusted return on capital and the Mean of Order P. These indicators offer a nuanced view of risk by capturing different aspects of tail behavior, which are critical in understanding potential extreme losses. These risk indicators are applied to analyze actuarial financial claims data, providing a robust framework for assessing financial stability and decision-making in the face of uncertainty. Full article

This study introduces the sine unit exponentiated half-logistic distribution, a novel extension of the unit exponentiated half-logistic distribution within the sine-G family. Using trigonometric transformations, the proposed distribution offers flexible density shapes for modeling asymmetric unit-interval data. We derive its statistical properties, including quantiles, moments, and stress–strength reliability, and estimate parameters via classical methods like maximum likelihood and Anderson–Darling. Simulations and real-world applications to fiber strength and burr datasets demonstrate the superior fit of the proposed distribution over competing models, highlighting its utility in reliability engineering and manufacturing. 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

We develop a novel family of distributions named the Marshall–Olkin type II exponentiated half-logistic–odd Burr X-G distribution. Several mathematical properties including linear representation of the density function, Rényi entropy, probability-weighted moments, and distribution of order statistics are obtained. Different estimation methods are employed to estimate the unknown parameters of the new distribution. A simulation study is conducted to assess the effectiveness of the estimation methods. A special model of the new distribution is used to show its usefulness in various disciplines. Full article

A novel odd generalized exponential Kumaraswamy–Weibull distribution is defined. This distribution is distinguished by its capacity to capture a wider class of hazard functions than the standard Weibull models, such as non-monotonic and bathtub-shaped hazards. This is an advancement in distribution theory because it provides a new simplified form of the distribution with a much more complicated behavior, which results in better statistical inference and detail in survival analysis and other related fields. Considerations on the identifiability of the proposed distribution are addressed, emphasizing the distinct contributions of its parameters and their roles in model behavior characterization. One real dataset from a survival experiment is considered, highlighting the practical implications of our distribution in the context of reliability. 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

In this paper, two different families are mixed: the exponentiated and new Topp–Leone-G families. This yields a new family, which we named the mixture of the exponentiated and new Topp–Leone-G family. Some statistical properties of the proposed family are obtained. Then, the mixture of two exponentiated new Topp–Leone inverse Weibull distribution is introduced as a sub-model from the mixture of exponentiated and new Topp–Leone-G family. Some related properties are studied, such as the quantile function, moments, moment generating function, and order statistics. Furthermore, the maximum likelihood and Bayes approaches are employed to estimate the unknown parameters, reliability and hazard rate functions of the mixture of exponentiated and new Topp–Leone inverse Weibull distribution. Bayes estimators are derived under both the symmetric squared error loss function and the asymmetric linear exponential loss function. The performance of maximum likelihood and Bayes estimators is evaluated through a Monte Carlo simulation. The applicability and flexibility of the MENTL-IW distribution are demonstrated by well-fitting two real-world engineering datasets. The results demonstrate the superior performance of the MENTL-IW distribution compared to other competing models. Full article

This paper proposes a system made up of four subsystems connected in sequence. The first and third subsystems each have one unit, the second has two, and the fourth has three. Every subsystem operates in parallel and is governed by the K-Out-of-n:G rule. Nonetheless, each subsystem needs at least one operational unit in order for the system to work. While a unit’s failure has an exponential distribution, repair is simulated using a general distribution and a distribution from the Gumbel–Hougaard family of copula. This study’s primary objective is to assess and contrast the system performance while our system is running under these two different repair policies. The problem is solved by combining the supplementary variable technique with Laplace transforms. We use reliability metrics to assess system performance. The second objective of this study is to present a reduction approach plan aimed at improving the overall reliability metrics of our system. 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 isotropic Cauchy distribution is a member of the central $\alpha$-stable family that plays a role in the set of heavy-tailed distributions similar to that of the Gaussian density among finite second-moment laws. Given a sequence of n observations, we are interested in characterizing the performance of Likelihood Ratio Tests, where two hypotheses are plausible for the observed quantities: either isotropic Cauchy or isotropic Gaussian. Under various setups, we show that the probability of error of such detectors is not always exponentially decaying with n, with the leading term in the exponent shown to be logarithmic instead, and we determine the constants in that leading term. Perhaps surprisingly, the optimal Bayesian probabilities of error are found to exhibit different asymptotic behaviors. Full article

The symmetric Kullback–Leibler centroid, also called the Jeffreys centroid, of a set of mutually absolutely continuous probability distributions on a measure space provides a notion of centrality which has proven useful in many tasks, including information retrieval, information fusion, and clustering. However, the Jeffreys centroid is not available in closed form for sets of categorical or multivariate normal distributions, two widely used statistical models, and thus needs to be approximated numerically in practice. In this paper, we first propose the new Jeffreys–Fisher–Rao center defined as the Fisher–Rao midpoint of the sided Kullback–Leibler centroids as a plug-in replacement of the Jeffreys centroid. This Jeffreys–Fisher–Rao center admits a generic formula for uni-parameter exponential family distributions and a closed-form formula for categorical and multivariate normal distributions; it matches exactly the Jeffreys centroid for same-mean normal distributions and is experimentally observed in practice to be close to the Jeffreys centroid. Second, we define a new type of inductive center generalizing the principle of the Gauss arithmetic–geometric double sequence mean for pairs of densities of any given exponential family. This new Gauss–Bregman center is shown experimentally to approximate very well the Jeffreys centroid and is suggested to be used as a replacement for the Jeffreys centroid when the Jeffreys–Fisher–Rao center is not available in closed form. Furthermore, this inductive center always converges and matches the Jeffreys centroid for sets of same-mean normal distributions. We report on our experiments, which first demonstrate how well the closed-form formula of the Jeffreys–Fisher–Rao center for categorical distributions approximates the costly numerical Jeffreys centroid, which relies on the Lambert W function, and second show the fast convergence of the Gauss–Bregman double sequences, which can approximate closely the Jeffreys centroid when truncated to a first few iterations. Finally, we conclude this work by reinterpreting these fast proxy Jeffreys–Fisher–Rao and Gauss–Bregman centers of Jeffreys centroids under the lens of dually flat spaces in information geometry. Full article

This paper presents a new methodology for generating continuous statistical distributions, integrating the exponentiated odds ratio within the framework of survival analysis. This new method enhances the flexibility and adaptability of distribution models to effectively address the complexities inherent in contemporary datasets. The core of this advancement is illustrated by introducing a particular subfamily, the “Type 2 Gumbel Weibull-G family of distributions”. We provide a comprehensive analysis of the mathematical properties of these distributions, including statistical properties such as density functions, moments, hazard rate and quantile functions, Rényi entropy, order statistics, and the concept of stochastic ordering. To test the robustness of our new model, we apply five distinct methods for parameter estimation. The practical applicability of the Type 2 Gumbel Weibull-G distributions is further supported through the analysis of three real-world datasets. These real-life applications illustrate the exceptional statistical precision of our distributions compared to existing models, thereby reinforcing their significant value in both theoretical and practical statistical applications. Full article