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The analysis and modeling of zero truncated count data is of primary interest in many elds such as engineering, public health, sociology, psychology, epidemiology. Therefore, in this article we have proposed a new and simple structure model, named a zero truncated discrete Lindley distribution. Thedistribution contains some submodels and represents a two-component mixture of a zero truncated geometric distribution and a zero truncated negative binomial distribution with certain parameters. Several properties of the distribution are obtained such as mean residual life function, probability generating function, factorial moments, negative moments, moments of residual life function, Bonferroni and Lorenz curves, estimation of parameters, Shannon and Renyi entropies, order statistics with the asymptotic distribution of their extremes and range, a characterization, stochastic ordering and stress-strength parameter. Moreover, the collective risk model is discussed by considering theproposed distribution as primary distribution and exponential and Erlang distributions as secondary ones. Test and evaluation statistics as well as three real data applications are considered to assess the peformance of the distribution among the most frequently zero truncated discrete probability models.

This paper introduces the bivariate bounded Gompertz–log-logistic (BBGLL) distribution, a bounded bivariate lifetime model built by coupling two bounded Gompertz–log-logistic marginals through a Clayton copula with an independent dependence parameter. The proposed model effectively describes positively dependent lifetimes within finite support and accommodates increasing, decreasing, and bathtub-shaped hazard rates. Analytical expressions for the survival functions, hazard rate functions, and joint moments are derived, while measures of association such as Kendall’s tau, Spearman’s rho, and tail-dependence coefficients characterize the dependence structure. Parameters are estimated via maximum likelihood, inference functions for margins (IFM), and semi-parametric methods, with performance assessed through Monte Carlo simulations. A real-life data application illustrates the practical relevance of the model, showing that the BBGLL distribution achieves a superior goodness-of-fit relative to existing bivariate alternatives, highlighting its practical usefulness.

Probabilistic streamflow models play a pivotal role in quantifying hydrological uncertainty and form the backbone of modern risk management strategies for flood and drought forecasting, water allocation planning, and the design of resilient infrastructure. Unlike deterministic approaches that yield single-point estimates, these models provide a spectrum of possible outcomes, enabling a more realistic assessment of extreme events and supporting informed, sustainable water resource decisions. By explicitly accounting for natural variability and uncertainty, probabilistic models promote transparent, robust, and equitable risk evaluations, helping decision-makers balance economic costs, societal benefits, and environmental protection for long-term sustainability. In this study, we introduce the bounded half-logistic distribution (BHLD), a novel heavy-tailed probability model constructed using the T–Y method for distribution generation, where T denotes a transformer distribution and Y represents a baseline generator. Although the BHLD is conceptually related to the Pareto and log-logistic families, it offers several distinctive advantages for streamflow modeling, including a flexible hazard rate that can be unimodal or monotonically decreasing, a finite lower bound, and closed-form expressions for key risk measures such as Value at Risk (VaR) and Tail Value at Risk (TVaR). The proposed distribution is defined on a lower-bounded domain, allowing it to realistically capture physical constraints inherent in flood processes, while a log-logistic-based tail structure provides the flexibility needed to model extreme hydrological events. Moreover, the BHLD is analytically characterized through a governing differential equation and further examined via its characteristic function and the maximum entropy principle, ensuring stable and efficient parameter estimation. It integrates a half-logistic generator with a log-logistic baseline, yielding a power-law tail decay governed by the parameter β, which is particularly effective for representing extreme flows. Fundamental properties, including the hazard rate function, moments, and entropy measures, are derived in closed form, and model parameters are estimated using the maximum likelihood method. Applied to four real streamflow data sets, the BHLD demonstrates superior performance over nine competing distributions in goodness-of-fit analyses, with notable improvements in tail representation. The model facilitates accurate computation of hydrological risk metrics such as VaR, TVaR, and tail variance, uncovering pronounced temporal variations in flood risk and establishing the BHLD as a powerful and reliable tool for streamflow modeling under changing environmental conditions.

This paper introduces Exponentiated Inverse Exponential Distribution (EIED), a novel probability model developed within the power inverse exponential distribution framework. A distinctive feature of EIED is its highly flexible hazard rate function, which can exhibit increasing, decreasing, and reverse bathtub (upside-down bathtub) shapes, making it suitable for modeling diverse lifetime phenomena in reliability engineering, survival analysis, and risk assessment. We derived comprehensive statistical properties of the distribution, including the reliability and hazard functions, moments, characteristic and quantile functions, moment generating function, mean deviations, Lorenz and Bonferroni curves, and various entropy measures. The identifiability of the model parameters was rigorously established, and maximum likelihood estimation was employed for parameter inference. Through extensive simulation studies, we demonstrate the robustness of the estimation procedure across different parameter configurations. The practical utility of EIED was validated through applications to real-world datasets, where it showed superior performance compared to existing distributions. The proposed model offers enhanced flexibility for modeling complex lifetime data with varying hazard patterns, particularly in scenarios involving early failure periods, wear-in phases, and wear-out behaviors.

Risk management is vital for financial institutions to evaluate and mitigate potential losses. Thunderstorm count modeling with risk analysis is used by various sectors, such as insurance and utility companies, to forecast storm recurrence, analyze risk, and estimate financial losses based on factors like wind speed, hail size, and tornado potential. This paper introduces a novel discrete distribution, the Beta-Gamma Discrete (BGD) distribution, designed for modeling count data that inherently excludes zero values. Developed through the compounding of a discrete gamma distribution with a beta distribution, the BGD offers significant flexibility in handling overdispersion and complex data characteristics. The study derives key statistical properties of the BGD, including its probability mass function, moments, hazard rate function, moment generating function, and mean residual life. A comprehensive characterization theorem is also established. The model’s practical utility is demonstrated through an application to thunderstorm event data from the Kennedy Space Center (KSC), where the frequency of thunderstorms per event is a critical operational concern. The performance of the BGD is thoroughly assessed against established zero-truncated models—namely, the Zero-Truncated Generalized Poisson (ZTGP), Size-Biased Negative Binomial (SBNB), and Zero-Truncated Generalized Negative Binomial (ZTGNB)—using evaluation criteria such as Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), Chi-square goodness-of-fit, and the Vuong test. The results consistently show that the BGD provides a superior and more accurate fit for the thunderstorm data, thus help NASA and other space agencies for establishing it as a robust and effective tool for modeling positive count data in meteorological and other applied contexts with risk analysis.

Distributions with bounded support show considerable sparsity over those with unbounded support, despite the fact that there are a number of real-world contexts where observations take values from a bounded range (proportions, percentages, and fractions are typical examples). For proportion modeling, a flexible family of two-parameter distribution functions associated with the exponential distribution is proposed here. The mathematical and statistical properties of the novel distribution are examined, including the quantiles, mode, moments, hazard rate function, and its characterization. The parameter estimation procedure using the maximum likelihood method is carried out, and applications to environmental and engineering data are also considered. To this end, various statistical tests are used, along with some other information criterion indicators to determine how well the model fits the data. The proposed model is found to be the most efficient plan in most cases for the datasets considered.

In this study, we introduce a novel three-parameter discrete probability distribution, defined by the parameters α, β, and λ, called the Generalized Nabla Geometric Distribution (GNGD). The proposed model is constructed through the discretization of a generalized mixture of exponential distributions. Among its parameters, β plays a significant role in governing the tail behavior and determining the shape of the hazard rate function. The adaptability of the GNGD is further highlighted by its capability to model under-dispersed, equi-dispersed, and over-dispersed data. Several important theoretical properties of the proposed distribution are explored, including the probability generating function, moments, and reliability characteristics, all obtained in explicit closed forms. In addition, the distribution is characterized, and the asymptotic behavior of its minimum and maximum order statistics is investigated. Parameter estimation is performed using the maximum likelihood estimation (MLE) approach, and an extensive simulation study is conducted to evaluate the efficiency and consistency of the estimators. To illustrate its practical usefulness, the GNGD is fitted to two real engineering data sets arising from repairable and non-repairable systems. The findings demonstrate that the proposed distribution provides an excellent fit to both data sets, achieving the smallest information loss criteria and the largest p-values among the competing goodness-of-fit measures considered.

We introduce the one-parameter bounded p-exponential distribution on (0, p+1), which includes the uniform model as a special case and converges pointwise to the exponential law as p→∞. Closed-form expressions are derived for the CDF and PDF, the survival function, an explicit increasing-failure-rate hazard function, the quantile function (enabling inversion-based simulation), moments, and entropy, along with a constructive scaled beta or Kumaraswamy representation. We also establish stochastic ordering with respect to p in stop-loss and increasing convex order, formalizing how dispersion varies with the parameter while preserving the mean scale. Inference is discussed under parameter-dependent support, a non-regular setting, and we develop and compare several estimation procedures, including a likelihood-based boundary MLE, a variance-matching method-of-moments estimator, and Bayesian estimation under a gamma prior implemented via numerical quadrature or MCMC. Monte Carlo simulation studies evaluate finite-sample performance and interval behavior, and two real-world applications in survival and reliability analysis illustrate competitive goodness-of-fit relative to standard benchmark models.

This paper proposes the lower-bounded Fréchet–log-logistic distribution (LFLD), a probability model designed for robust flood frequency analysis (FFA). The LFLD addresses key limitations of traditional distributions (e.g., generalized extreme value (GEV) and log-Pearson Type III (LP3)) by combining bounded support (α

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