Search Results (467)

Recent developments in discrete probability models play a crucial role in reliability and survival analysis when lifetimes are recorded as counts. Motivated by this need, we introduce the discrete ZLindley (DZL) distribution, a novel discretization of the continuous ZL law. Constructed using a survival-function approach, the DZL retains the analytical tractability of its continuous parent while simultaneously exhibiting a monotonically decreasing probability mass function and a strictly increasing hazard rate—properties that are rarely achieved together in existing discrete models. We derive key statistical properties of the proposed distribution, including moments, quantiles, order statistics, and reliability indices such as stress–strength reliability and the mean residual life. These results demonstrate the DZL’s flexibility in modeling skewness, over-dispersion, and heavy-tailed behavior. For statistical inference, we develop maximum likelihood and symmetric Bayesian estimation procedures under censored sampling schemes, supported by asymptotic approximations, bootstrap methods, and Markov chain Monte Carlo techniques. Monte Carlo simulation studies confirm the robustness and efficiency of the Bayesian estimators, particularly under informative prior specifications. The practical applicability of the DZL is illustrated using two real datasets: failure times (in hours) of 18 electronic systems and remission durations (in weeks) of 20 leukemia patients. In both cases, the DZL provides substantially better fits than nine established discrete distributions. By combining structural simplicity, inferential flexibility, and strong empirical performance, the DZL distribution advances discrete reliability theory and offers a versatile tool for contemporary statistical modeling. Full article

Rainfall infiltration is one of the main factors inducing slope instability, while the spatial heterogeneity and uncertainty of soil parameters have profound impacts on slope response characteristics and stability evolution. Traditional deterministic analysis methods struggle to reveal the dynamic risk evolution process of the system under heavy rainfall. Therefore, this paper proposes an uncertainty analysis framework combining Karhunen–Loève Expansion (KLE) random field theory, Polynomial Chaos Kriging (PCK) surrogate modeling, and Monte Carlo simulation to efficiently quantify the probabilistic characteristics and spatial risks of rainfall-induced slope instability. First, for key strength parameters such as cohesion and internal friction angle, a two-dimensional random field with spatial correlation is constructed to realistically depict the regional variability of soil mechanical properties. Second, a PCK surrogate model optimized by the LARS algorithm is developed to achieve high-precision replacement of finite element calculation results. Then, large-scale Monte Carlo simulations are conducted based on the surrogate model to obtain the probability distribution characteristics of slope safety factors and potential instability areas at different times. The research results show that the slope enters the most unstable stage during the middle of rainfall (36–54 h), with severe system response fluctuations and highly concentrated instability risks. Deterministic analysis generally overestimates slope safety and ignores extreme responses in tail samples. The proposed method can effectively identify the multi-source uncertainty effects of slope systems, providing theoretical support and technical pathways for risk early warning, zoning design, and protection optimization of slope engineering during rainfall periods. Full article

We study waiting times in a queue with active management and correlated job/packet sizes, which induce correlated service times. In the transient case, formulae for the distribution tail, the probability density, and the expected virtual waiting time at any time t, are found. Then, grounded on these results, stationary versions of the probability density and the expected value are obtained. The correlation of service times resulting from correlated job sizes is modeled through an MSP (Markovian service process). Theoretical results are reinforced by numerical examples, in which we examine the impact of symmetric positive and negative correlation of service times, and the impact of symmetric weak and strong active management, on transient and stationary waiting times. We also compare the effects of these factors on the waiting time with their effects on the queue length. In these examples, we can see a surprisingly large expected virtual waiting time, much greater than the product of the expected service time and the queue length. This effect is observed for both weak and strong management functions when the correlation is positive, but it vanishes when a symmetric negative correlation is applied. We also observe a weaker effect of active management on virtual waiting times than that of service time correlation, as well as a weaker impact of active management on virtual waiting time densities than on queue length distributions. Full article

This research explores the application of extreme value theory in modelling and quantifying tail risks across different economic equity markets, with focus on the Nairobi Securities Exchange (NSE20), the South African Equity Market (FTSE/JSE Top40) and the US Equity Index (S&P500). The study aims to recommend the most suitable probability distribution between the Generalised Extreme Value Distribution (GEVD) and the Generalised Pareto Distribution (GPD) and to assess the associated tail risk using the value-at-risk and expected shortfall. To address volatility clustering, four generalised autoregressive conditional heteroscedasticity (GARCH) models (standard GARCH, exponential GARCH, threshold-GARCH and APARCH (asymmetric power ARCH)) are first applied to returns before implementing the peaks-over-threshold and block maxima methods on standardised residuals. For each equity index, the probability models were ranked based on goodness-of-fit and accuracy using a combination of graphical and numerical methods as well as the comparison of empirical and theoretical risk measures. Beyond its technical contributions, this study has broader implications for building sustainable and resilient financial systems. The results indicate that, for the GEVD, the maxima and minima returns of block size 21 yield the best fit for all indices. For GPD, Hill’s plot is the preferred threshold selection method across all indices due to higher exceedances. A final comparison between GEVD and GPD is conducted to estimate tail risk for each index, and it is observed that GPD consistently outperforms GEVD regardless of market classification. Full article

This paper proposes a computationally efficient framework for estimating first-passage probabilities of nonlinear structures under stochastic seismic excitations. The methodology integrates Optimal Latinized Partially Stratified Sampling (OLPSS) with the Random Function Spectral Representation Method (RFSRM) to generate a minimal yet optimal set of samples in the low-dimensional input space. Each sample corresponds to a representative nonstationary ground motion time history, which is then used to drive nonlinear dynamic analyses. The extreme values of the structural responses are extracted, and their distribution tails are accurately modeled using the Shifted Generalized Lognormal Distribution (SGLD), whose parameters are efficiently estimated via an extrapolation method. This allows for the construction of the probability density function (PDF) and cumulative distribution function (CDF) of the extreme responses, from which the failure probabilities and reliability indices are calculated. The proposed framework is rigorously validated against the Monte Carlo simulation (MCS) benchmarks using two illustrative examples, including a nonlinear single-degree-of-freedom (SDOF) system and a three-story shear building model. The results demonstrate that the proposed method achieves excellent accuracy in estimating failure probabilities and reliability indices, while significantly reducing the number of required simulations and thereby confirming its high efficiency and accuracy for rapid performance-based seismic assessment. Full article

This study aims to fill the existing gap in laser detection research, particularly regarding how the waveform of outgoing laser pulses affects detection performance. Based on the mechanism of light cone beam expansion, this study emits three different laser pulse signals to detect short-range targets. A theoretical model for short-range ranging of these lasers is established, and the effects of emission power, divergence angle, and equivalent root mean square noise voltage on circumferential detection accuracy are simulated and experimentally measured. As emission power decreases, both echo amplitude and detection accuracy decline for all three pulsed lasers. Additionally, except for the inverted parabolic function, both echo amplitude and detection accuracy decrease with reduced divergence angle. An increase in equivalent root mean square noise voltage broadens the half-width of the probability density distribution for pulsed laser detection. The mean central position deviation between the ideal and measured detection probability density distributions of the heavy-tailed function laser pulses shows the best performance and the highest fidelity, which are +0.01 m, +0.05 m, and +0.02 m, respectively, which is of great significance for the development of laser detection technology. Full article

We investigate the fundamental trade-off between entropy and the Gini index within income distributions, employing a stochastic framework to expose deficiencies in conventional inequality metrics. Anchored in the principle of maximum entropy (ME), we position entropy as a key marker of societal robustness, while the Gini index, identical to the (second-order) K-spread coefficient, captures spread but neglects dynamics in distribution tails. We recommend supplanting Lorenz profiles with simpler graphs such as the odds and probability density functions, and a core set of numerical indicators (K-spread $K_2/\mu$, standardized entropy ${\Phi }_{\mu }$, and upper and lower tail indices, $\xi , \zeta$) for deeper diagnostics. This approach fuses ME into disparity evaluation, highlighting a path to harmonize fairness with structural endurance. Drawing from percentile records in the World Income Inequality Database from 1947 to 2023, we fit flexible models (Pareto–Burr–Feller, Dagum) and extract K-moments and tail indices. The results unveil a concave frontier: moderate Gini reductions have little effect on entropy, but aggressive equalization incurs steep stability costs. Country-level analyses (Argentina, Brazil, South Africa, Bulgaria) link entropy declines to political ruptures, positioning low entropy as a precursor to instability. On the other hand, analyses based on the core set of indicators for present-day geopolitical powers show that they are positioned in a high stability area. Full article

Swine Inflammation and Necrosis Syndrome (SINS) is a highly prevalent, predominantly endogenous condition that compromises tissue integrity and animal welfare across different life stages in pigs. Increasing evidence suggests that early-life SINS lesions may predispose pigs to tail damage later in life; however, longitudinal data remain scarce. This study investigated the association between SINS-related clinical signs in suckling piglets and weaners and subsequent tail integrity during fattening and at slaughter. In a longitudinal study, 352 piglets from two Italian farms producing Parma ham were followed from the suckling phase to slaughter. Although SINS signs were generally mild, pigs affected during the weaner phase showed a 3.5-fold increased risk of developing short tails during fattening. Furthermore, the probability of reduced tail length at slaughter increased from 33.5% to 65.8% in pigs with a history of SINS. Early-life SINS was significantly associated with impaired tail integrity both at the onset of fattening and at slaughter. These new findings highlight endogenous inflammation and necrosis in early life as important yet underrecognized welfare risk factors and suggest that SINS can be utilised as a point of care and early preventive strategies may substantially improve tail integrity and welfare outcomes at slaughter. Full article

We establish precise exponential tail estimates for lacunary trigonometric sums of the form $f_N\left(x\right)={\sum }_{k=1}^N{c}_k\cos \left(2\pi n_{k}x\right)$, under the Hadamard gap condition. Using cumulant expansions and moment-generating function techniques, we obtain non-asymptotic upper bounds for the tail probabilities, including third-order corrections that refine the classical central limit theorem estimates. Furthermore, several examples illustrate these bounds for various choices of coefficients, highlighting the transition from subgaussian to stretched-exponential tail behavior. Full article

We introduce the Symmetric-$\mathbb{Z}$ (Sy-$\mathbb{Z}$) family, a unified class of symmetric discrete distributions on the integers obtained by multiplying a three-point symmetric sign variable by an independent non-negative integer-valued magnitude. This sign-magnitude construction yields interpretable, zero-centered models with tunable mass at zero and dispersion balanced across signs, making them suitable for outcomes, such as differences of counts or discretized return increments. We derive general distributional properties, including closed-form expressions for the probability mass and cumulative distribution functions, bilateral generating functions, and even moments, and show that the tail behavior is inherited from the magnitude component. A characterization by symmetry and sign–magnitude independence is established and a distinctive operational feature is proved: for independent members of the family, the sum and the difference have the same distribution. As a central example, we study the symmetric Poisson model, providing measures of skewness, kurtosis, and entropy, together with estimation via the method of moments and maximum likelihood. Simulation studies assess finite-sample performance of the estimators, and applications to datasets from finance and education show improved goodness-of-fit relative to established integer-valued competitors. Overall, the Sy-$\mathbb{Z}$ framework offers a mathematically tractable and interpretable basis for modeling symmetric integer-valued outcomes across diverse domains. Full article

Tailings dams, critical for storing mine waste and water, must maintain stability and functionality throughout their lifespan. Their design and risk assessment are complicated by significant uncertainties stemming from multivariable parameters, including material properties, loading conditions, and operational decisions. Traditional dam design and risk assessment procedures often rely on first-order probabilistic approaches, which fail to capture the complex, multi-layered nature of these uncertainties fully. This paper reviews the current tailings dam design practice and proposes the application of Bayesian networks (BNs) to analyse the epistemic and aleatory uncertainty inherent in tailings dam design parameters and risk assessment. By representing these uncertainties explicitly, BNs can facilitate more robust and targeted design strategies. The proposed approach involves several key steps, including parameterisation—design input variable probability density function and uncertainty, knowledge elicitation, and model assessment and integration. This methodology provides a sophisticated and comprehensive approach to accounting for the full spectrum of uncertainties, thereby enhancing the reliability of tailings dam designs and risk management decisions. Full article

The rapid growth of distributed photovoltaic (PV) resources is transforming distribution networks into active systems with highly variable net loads, while the rising frequency and severity of extreme weather events is increasing outage risk and restoration challenges. In this context, utilities require reliability assessment tools that jointly represent operational variability and climate-driven stressors beyond stationary assumptions. This paper presents a weather-aware probabilistic framework to quantify the reliability of active distribution networks with high PV penetration. The approach synthesizes realistic residential demand and PV time series at 15-min resolution, models extreme weather as a low-probability/high-impact escalation of component failure rates and restoration uncertainty, and computes IEEE Std 1366–2022 indices (SAIFI, SAIDI, ENS) through Monte Carlo simulation. The methodology is validated on a modified IEEE 33-bus feeder with parameter values representative of urban/suburban overhead networks. Compared with classical reliability modeling, the proposed framework captures in a unified pipeline the joint effects of load/PV stochasticity, weather-dependent failure escalation, and repair-time dispersion, providing a consistent statistical interpretation supported by kernel density estimation and convergence diagnostics. The results show that (i) extreme weather shifts the distributions of SAIFI, SAIDI and ENS to the right and thickens upper tails (higher exceedance probabilities); (ii) PV penetration yields a non-monotonic response with measurable improvements up to intermediate levels and saturation/partial degradation at very high penetrations; and (iii) compound risk is nonlinear, as the mean ENS surface over $(r_{\mathrm{PV}},P_{\mathrm{ext}})$ exhibits a valley at moderate PV and a ridge for large storm probability. A tornado analysis identifies the base failure rate, storm escalation factor and storm exposure as dominant drivers, in line with resilience literature. Overall, the framework provides an auditable, scenario-based tool to co-design DER hosting and resilience investments. 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 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

This paper introduces novel non-parametric entropy-based evidence functions and associated test statistics for assessing the dilation order of probability distributions constructed from cumulative residual entropy and cumulative entropy. The proposed evidence functions are explicitly tuned to questions about distributional variability and stochastic ordering, rather than global model fit, and are developed within a rigorous evidential framework. Their asymptotic distributions are established, providing a solid foundation for large-sample inference. Beyond their theoretical appeal, these procedures act as effective entropy-driven tools for quantifying statistical evidence, offering a compelling non-parametric alternative to traditional approaches, such as Kullback–Leibler discrepancies. Comprehensive Monte Carlo simulations highlight their robustness and consistently high power across a wide range of distributional scenarios, including heavy-tailed models, where conventional methods often perform poorly. A real-data example further illustrates their practical utility, showing how cumulative entropies can provide sharper statistical evidence and clarify stochastic comparisons in applied settings. Altogether, these results advance the theoretical foundation of evidential statistics and open avenues for applying cumulative entropies to broader classes of stochastic inference problems. Full article