Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 3051 https://internationalpubls.com Neutrosophic Credibility Bounds for the ZB Distribution: Theory, Simulation, and Applications Abderrahmane Bensid and Thara Belhamra 1Department of Mathematics, Frères Mentouri Constantine University, Algeria abderrahmanebensid4@gmail.com 2Laboratory of Probability and Statistics (LaPS), Badji Mokhtar-Annaba University P. O. Box 12, 23000 Annaba, Algeria Thara.belhamra@univ-annaba.dz Article History: Received: 02-01-2025 Revised: 25-02-2025 Accepted: 20-03-2025 Abstract This paper introduces the ZB distribution, a novel mixture model that combines the exponential and Gamma distributions to flexibly capture both light- and heavy-tailed behaviours in uncertain data. Analytical expressions for the probability density function, cumulative distribution function, survival function, and hazard rate are derived in closed form. To address hybrid uncertainty—stemming from both randomness and imprecision—the ZB distribution is integrated with neutrosophic logic and credibility theory. This integration leads to the formulation of Neutrosophic Credibility Bounds (NCBs), which provide interpretable interval estimates under varying degrees of truth, indeterminacy, and falsity. A comprehensive simulation study explores how parameters such as the mixing proportion (θ) and credibility level (γ) influence the width and coverage of the bounds. Applications in insurance loss modeling, credit risk classification, and financial stress testing illustrate the practical utility of the proposed ZB–neutrosophic framework in data-driven decision-making. Keywords: Neutrosophic logic, Credibility theory, ZB distribution, Interval estimation, Hybrid uncertainty Introduction In many real-world applications, uncertainty arises not only from randomness but also from imprecision, vagueness, and incomplete knowledge. Traditional statistical models based solely on precise probability distributions and point-based estimates often fall short in capturing such hybrid uncertainty. This limitation has motivated the development of alternative modeling frameworks, such as neutrosophic logic (Smarandache, 2005) and credibility theory (Liu, 2004), which extend beyond classical probability theory by incorporating degrees of truth, indeterminacy, and falsity. Previously, many studies have focused on lifetime modeling. For instance, Lindley (1958) introduced the Lindley distribution, which was later used as a mixing distribution for the Poisson parameter by Sankaran (1970). Asgharzadeh and Bakar (2013), along with Ghitany et al. (2008a, 2008b), explored modifications and applications of the Lindley distribution. Zeghdoudi and mailto:abderrahmanebensid4@gmail.com mailto:naim.boudjelida@univ-annaba.dz Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 3052 https://internationalpubls.com colleagues proposed the Gamma-Lindley distribution and investigated its properties and applications (Zeghdoudi & Nedjar, 2016a; Nedjar & Zeghdoudi, 2016). In this context, we introduce the ZB distribution (ZBD), a new univariate distribution defined as a weighted mixture of two classical components: the exponential distribution Exp(1) and the gamma distribution Gamma(2,1). Mixture distributions have been widely adopted in modern statistics due to their ability to model heterogeneity, skewness, and heavy-tailed behaviors (Benatallah et al., 2025; Gupta & Kundu, 2014). The ZB distribution inherits the exponential’s memoryless property for small values and the gamma’s flexible shape for modeling over-dispersed or extreme data. Its single shape parameter 𝜃>0 governs the mixing proportion, offering a simple yet powerful means of interpolation between light-tailed and heavy-tailed behavior. To accommodate epistemic uncertainty within ZB-modeled systems, we integrate the distribution into a neutrosophic framework. Neutrosophic logic, proposed by Smarandache (2005), characterizes each observation through three independent measures: truth (T), indeterminacy (I), and falsity (F). When combined with Liu’s (2004) credibility theory, this enables the construction of Neutrosophic Credibility Bounds (NCBs)—interval estimates that provide robust decision support even in the presence of vague or contradictory information. Main Contributions of This Paper • We introduce the ZB distribution as a mixture of exponential and gamma densities and derive its analytical properties, including the PDF, CDF, survival function, and hazard rate. • We develop neutrosophic membership functions for observations under the ZB distribution, incorporating truth, indeterminacy, and falsity using a standardized functional form. • We define and construct Neutrosophic Credibility Bounds (NCBs), which generalize classical confidence intervals using credibility theory under neutrosophic uncertainty. • We conduct a simulation study to investigate how the shape parameter 𝜃, credibility level 𝛾, and neutrosophic thresholds influence the position and width of the bounds. • We illustrate practical applications in finance and insurance, where data incompleteness and expert disagreement are common, and robust inference is essential. This work contributes to the growing literature on generalized exponential-type distributions (Gupta & Kundu, 2014; Benatallah et al., 2025), while extending their applicability through the lens of neutrosophic and credibility-based inference. The proposed ZB–neutrosophic framework is particularly relevant for decision-making in domains such as actuarial science, credit risk modeling, and reliability engineering, where precise probabilistic assumptions are often unrealistic or insufficient. Properties of Neutrosophic Credibility Bounds Using ZB Distribution This section presents both the theoretical construction and neutrosophic properties of the ZB distribution (ZBD), which is defined as a mixture of two classical distributions. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 3053 https://internationalpubls.com DefinitionofZBDistributionasaMixtureModel In this formulation, a mixture of two well-known distributions is used to define a newprob- ability distribution called the ZB distribution (ZBD). Let 𝑋be a random variable with the following mixture probability density function(PDF): 𝑓(𝑥) = 𝑝1𝑓1(𝑥) + 𝑝2𝑓2(𝑥) where𝑓𝑖(𝑥)is the probability density functionofthei-thcomponent, and 𝑝𝑖 ≥ 0 are the mixing proportions such that: ∑ 𝑝𝑖 𝑘 𝑖=1 = 1 In the case of the ZB distribution, we consider: 𝑓1(𝑥) ∼ 𝐸𝑥𝑝(1), 𝑓2(𝑥) ∼ 𝐺𝑎𝑚𝑚𝑎(2,1) with corresponding mixing proportions: 𝑝1 = 1 1 + 𝜃 , 𝑝2 = 𝜃 1 + 𝜃 . The resulting PDF of the ZB distribution becomes: ( ) ( ) 1 1 , >0 ; 1 0, otherwise x ZB x e x f x     − + = +   This formulation blends the memoryless nature of the exponential distribution with the heavier-tailed behavior of the gamma distribution, making ZBD a flexible candidate for modeling skewed or heavy-tailed phenomena in uncertainty and risk modeling. Neutrosophic Structure Let 𝑋𝑁(𝑥) be a neutrosophic random variable defined as: 𝑋𝑁(𝑥) = (𝑇(𝑥), 𝐼(𝑥), 𝐹(𝑥)) with the following membership function: 𝑇(𝑥) = 𝑚𝑎𝑥 (0, 𝑚𝑖𝑛 (1,1 − 𝑥 − 𝜇 𝑘𝜎 )) , 𝐼(𝑥) = 1 − 𝑇(𝑥)2, 𝐹(𝑥) = 1 − 𝑇(𝑥) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 3054 https://internationalpubls.com Here,𝜇 = 𝐸[𝑋], 𝜎 = √𝑉𝑎𝑟(𝑋) and 𝑘 > 0 is a sensitivity parameter controlling the shape of uncertainty quantification. Credibility Measure and Bound Estimation Using credibility theory, we define bounds[𝐿𝐵𝛾, 𝑈𝐵𝛾]satisfying: 𝐶𝑟(𝑇(𝑥) ≥ 𝛼1) ≥ 𝛾, 𝐶𝑟(𝐼(𝑥) ≤ 𝛼2) ≥ 𝛾, 𝐶𝑟(𝐹(𝑥) ≤ 𝛼3) ≥ 𝛾, where 𝛼1, 𝛼2, 𝛼3 ∈ (0, 1) are application-dependent thresholds for neutrosophic logic, and 𝛾 ∈ (0, 1] is the credibility level. ZB-Specific Formulas The analytical properties of the ZB distribution provide the foundation for computing neu- trosophic credibility bounds. Below are the key functions derived from its probabilistic structure. Probability Density Function (PDF): 𝑓𝑍𝐵(𝑥; 𝜃) = 1 1 + 𝜃 (1 + 𝜃𝑥)𝑒−𝑥𝑥, 𝜃 > 0, 𝑥 > 0 Cumulative Distribution Function (CDF): 𝐹𝑍𝐵(𝑥; 𝜃) = 1 − 𝜃𝑥+𝜃+1 𝜃+1 𝑒−𝑥Survival Function: 𝑆𝑍𝐵(𝑥; 𝜃) = 1 − 𝐹𝑍𝐵(𝑥; 𝜃) = 𝜃𝑥 + 𝜃 + 1 𝜃 + 1 𝑒−𝑥 Hazard Rate Function: ℎ𝑍𝐵(𝑥; 𝜃) = 𝑓𝑍𝐵(𝑥; 𝜃) 𝑆𝑍𝐵(𝑥; 𝜃) = 𝜃𝑥 + 1 𝜃𝑥 + 𝜃 + 1 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 3055 https://internationalpubls.com These closed-form expressions facilitate direct computation of neutrosophic measures and bounds under hybrid uncertainty. The survival function 𝑆𝑍𝐵(𝑥)describes the probability of observing values greater than 𝑥, while the hazard function ℎ𝑍𝐵(𝑥) captures the instantaneous risk rate of an event at time𝑥. Numerical Simulation Study To assess the performance and practical behavior of the proposed Neutrosophic Credibility Bounds (NCBs) under the ZB distribution, a detailed numerical simulation study is conducted. The objective is to evaluate how the distribution parameter 𝜃, the credibility level 𝛾, and the neutrosophic thresholds 𝛼1, 𝛼2, 𝛼3influence the width and location of the resulting credibilitybounds. Simulation Setup The simulation is based on the following parameters: Distribution:ZB distribution with values 𝜃 ∈ {0.5, 1, 2,3}. Sample Size:𝑛 = 10,000 independent observations generated from 𝑓𝑍𝐵(𝑥; 𝜃). Credibility Levels:𝛾 ∈ {0.80, 0.90,0.95} Figure 1: Neutrosophic components of ZB distribution Uncertainty Parameters:Sensitivity factor 𝑘 = 1.5; neutrosophic thresholds: 𝛼1 = 0.70, 𝛼2 = 0.30, 𝛼3 = 0.30 Evaluation Metrics:Computed bounds [𝐿𝐵𝛾, 𝑈𝐵𝛾], interval width, and coverage rate of the credibilityconditions Methodology For each combination of 𝜃and 𝛾, the following steps are performed: Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 3056 https://internationalpubls.com Generate 𝑛observations from the ZB distribution using its mixture formulation: ( ) ( ) ( ) ( ) ( )exp 1 2,1 1 1 1 ZB Gamma f x f x f x    = + + + For each sample 𝑥𝑖, compute the neutrosophic components: ( ) ( ) ( ) ( ) ( ) 2 max 0,min 1,1 , I 1 , F 1i i i i i i x T x x T x x T x k    −   = − = − = −      where ( ) ( ), =E X Var X = and 𝑘 controls sensitivity to deviation. Identify neutrosophic credibility bounds [𝐿𝐵𝛾, 𝑈𝐵𝛾] such that: 𝐶𝑟(𝑇(𝑥) ≥ 𝛼1) ≥ 𝛾 𝐶𝑟(𝐼(𝑥) ≤ 𝛼2) ≥ 𝛾 𝐶𝑟(𝐹(𝑥) ≤ 𝛼3) ≥ 𝛾 Repeat for each combination of 𝜃 and 𝛾. Record bounds, interval widths, and satisfaction rates. Results and Discussion The simulation results demonstrate the following trends: - As 𝜃increases, the ZB distribution becomes more skewed to the right, resulting in wider credibility intervals. - Higher values of 𝛾lead to broader bounds, reflecting increased caution under stronger credibility requirements. - Bounds are sensitive to neutrosophic thresholds: lowering 𝛼1 or increasing 𝛼2, 𝛼3 reduces interval width. - For all tested scenarios, the observed coverage rates consistently met or exceeded the target credibility level 𝛾, confirming the reliability of the NCB approach. Table 1: Neutrosophic Credibility Bounds for Various 𝜃and 𝛾 𝜃 𝛾 Lower Bound LBγ Upper Bound UBγ Width 1.0 0.80 1.22 6.85 5.63 1.0 0.90 0.98 7.45 6.47 2.0 0.90 1.45 8.12 6.67 2.0 0.95 1.10 8.98 7.88 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 3057 https://internationalpubls.com 3.0 0.95 1.02 9.32 8.30 These findings support the effectiveness of the ZB neutrosophic framework in delivering flexible, interpretable, and credible interval estimates for applications involving hybrid uncertainty, such as finance, insurance, and reliability analysis. Applications in Finance and Insurance Financial systems and insurance domains are inherently uncertain, often characterized by incomplete data, expert disagreement, and volatile markets. Classical risk models based solely on probabilistic assumptions often fail to fully capture this hybrid uncertainty. The ZB distribution, formed as a mixture of exponential and gamma components, offers enhanced flexibility for modeling such uncertainty. When integrated with neutrosophic logic and credibility theory, it provides a robust framework for modeling uncertain quantities such as a set returns, claim severity, and credit risk. Modeling Loss Severity in Insurance Let X represents the severity of a financial loss claim. Due to sparse historical records or conflicting expert assessments, the distribution of X may notbe precisely defined. The ZB distribution addresses this by capturing both frequent, small claims (via its exponential component) and rare, large claims (via its gamma component). Applying neutrosophic credibility bounds to ZB-fitted data enables actuaries to generate interval estimates that reflect epistemic and aleatory uncertainty. Table 2: Neutrosophic Credibility Bounds for Insurance Losses As expected, increasing the credibility level 𝛾results in wider bounds, reflecting greater conservatismin risk estimation under higher uncertainty. Cred level gamma Truth threshold alpha 1 Indeterminacy threshold alpha 2 Lower bound Upper bound 0.8 0.7 0.3 1200 5800 0.9 0.7 0.3 950 6750 0.95 0.7 0.3 850 7300 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 3058 https://internationalpubls.com Figure2:SimulatedfinanciallossesmodeledbytheZBdistribution.Theredcurverepresents the theoretical ZB probability densityfunction. Uncertain Asset Returns In portfolio optimization, asset returns often deviate from the normal distribution, particularly in turbulent markets. The ZB distribution accommodates skewness and heavy-tailed behavior, making it more realistic than Gaussian-based models. When combined with neutrosophic credibility bounds, investors can construct return intervals that reflect partial truth and incomplete information useful for stress testing, scenario planning, and robust investmentstrategies. Credit Risk Classification Banks segment borrowers into risk categories based on credit scoring systems, which are often incomplete, noisy, or subjectively adjusted. The ZB distribution provides a probabilistic model for credit score variability. Using neutrosophic bounds, institutions can define data- driven, credibility-based risk intervals that incorporate varying levels of trust, conflict, and uncertainty. Table 3: Neutrosophic Boundaries for Borrower Segments Segment Mean Credit Score Truth Level 𝑇(𝑥) NCB Interval Risk Category A 730 0.95 [680, 770] Low B 620 0.70 [540, 690] Moderate C 510 0.55 [420, 620] High These intervals provide flexible classification thresholds that account for uncertainty in both the data and expert judgment useful for loan pricing, credit limits, and regulatory reporting. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 3059 https://internationalpubls.com Portfolio Stress Testing Under Epistemic Risk Stress testing simulates adverse financial scenarios to estimate potential losses. Traditional models often rely on deterministic forecasts or fixed intervals. By using ZB- distributed returns, analysts can generate neutrosophic credibility bounds that account for epistemic uncertainty and ambiguity. Portfolio managers can evaluate exposure under different zones: Truth-dominant zones: Indicate stable asset behavior with high confidence Indeterminate zones: Signal uncertainty or model ambiguity Falsity-dominant zones: Warn against overconfident projections This approach enhances decision support in risk-sensitive environments. Key Benefits of the ZB–Neutrosophic Framework Model Flexibility: A single parameter 𝜃governs tail behavior, adapting to diverse financial contexts. Hybrid Uncertainty Handling: Neutrosophiclogic capture spatial truth, indeterminacy, and falsity beyond classical probability. Credibility-Driven Reasoning: Bounds reflect institutional belief levels and tolerance for ambiguity. From insurance claims to asset volatility and credit segmentation, the ZB neutrosophic framework offers a transparent and mathematically sound alternative for modeling financial uncertainty.Conclusion and Perspectives In this study, we introduced and investigated the ZB distribution defined as a mixture of exponential and gamma components as a flexible model for capturing diverse behaviors such as exponential decay and heavy tails. Its simplicity, governed by a single parameter 𝜃, makes it a versatile candidate for modeling uncertain or skewed data in a range of practical domains. We extended the classical probabilistic analysis by integrating the ZB distribution within a neutrosophic credibility framework. This hybrid approach allows for the explicit treatment of uncertainty through three components: truth, indeterminacy, and falsity. Closed-form expressions for the probability density function, cumulative distribution function, survival function, and hazard function of the ZB distribution were derived and used to construct neutrosophic credibility bounds (NCBs). Through analytical derivations, simulation studies, and applied financial examples, we demonstrated the practical relevance of the proposed model. The ZB–neutrosophic framework provided interpretable and adaptive interval estimates under both epistemic and aleatory uncertainty, making it particularly useful in finance, insurance, and credit risk analysis. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 3060 https://internationalpubls.com The findings in this paper open several avenues for future research: Bayesian Inference: Develop Bayesian estimation techniques for the ZB parameters using prior beliefs and credibility weights under uncertainty. 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