EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6376 ISSN 1307-5543 – ejpam.com Published by New York Business Global A Novel U-Statistic Test for Exponentiality Against EBUCL Reliability Class and Applied to Complete and Censored Data Across Various Risk Profiles Walid B. H. Etman1, Mohamed F. Abouelenein2, Mohamed S. Eliwa3,4, Mahmoud El-Morshedy5,6,∗, Noura Roushdy2, Rashad M. EL-Sagheer7,8 1 Faculty of Computer and Artificial Intelligence, Modern University for Technology and Information, Cairo, Egypt 2 Department of Insurance and Risk Management, College of Business, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11432, Riyadh, Saudi Arabia 3 Department of Statistics and Operations Research, College of Science, Qassim University, Saudi Arabia 4 Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt. 5 Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia 6 Department of Statistics and Computer Science, Faculty of Science, Mansoura University, Mansoura 35516, Egypt 7 Mathematics Department, Faculty of Science, Al-Azhar University, Naser city 11884, Cairo, Egypt 8 High Institute of Computer and Management Information System, First Statement, New Cairo 11865, Cairo, Egypt Abstract. Statistical testing plays a pivotal role in enabling researchers to draw sound conclu- sions from data. Nonparametric tests, in particular, are highly valuable due to their flexibility in handling various data sets without requiring assumptions about the underlying distribution. In response to the growing need for robust testing procedures, this study introduces a new class of life distributions known as exponential better than used in convex Laplace transform order (EBUCL). A novel U-statistic-based test is developed to evaluate exponentiality against this class. The asymptotic properties of the proposed test are thoroughly examined, and critical values for sample sizes ranging from 5 to 50 are reported. A detailed simulation study evaluates the test’s power under commonly encountered reliability models. Moreover, Pitman’s asymptotic efficiency is calculated and compared with that of existing methods. The study also extends the methodology to handle right-censored data. Finally, the practical utility of the proposed test is demonstrated through applications to several real-world data sets from diverse fields. 2020 Mathematics Subject Classifications: 62N01, 62N02, 62N05, 62C07, 62G99 Key Words and Phrases: Nonparametric Statistics, Ageing Classifications in Reliability, Failure Analysis, Asymptotic Test Efficiency, Decision Support Systems ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6376 Email address: m.elmorshedy@psau.edu.sa (Mahmoud El-Morshedy) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) M. S. Eliwa et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6376 2 of 24 1. Introduction The integration of sustainability-driven analysis with censored data within the frame- work of reliability aging classes of life distributions represents a critical advancement in the understanding of product durability and lifespan. This integration not only enhances our ability to assess product performance over time but also plays a crucial role in in- forming decision-making related to resource management, environmental impact, and the evolution of sustainable methodologies. By analyzing censored data, which is often preva- lent in real-world scenarios where full lifetime information is unavailable, researchers can derive valuable insights into the reliability and resilience of products. These insights are fundamental for evaluating how products perform under varying conditions and contribute to their overall longevity. Moreover, this integrated analytical approach equips decision- makers with the tools needed to assess environmental footprints and develop strategies to mitigate adverse impacts. By examining the relationships between product durability, reliability, and sustainability, this method supports the creation of eco-conscious strategies and promotes the adoption of sustainable practices across industries. Through compre- hensive statistical analyses, this approach not only provides a clearer picture of product reliability but also lays the foundation for the development of robust sustainability frame- works. These frameworks are essential for improving resource efficiency, reducing waste, and advancing responsible environmental stewardship. Reliability, as a concept, is central to ensuring that products and systems consistently meet performance standards over time. It is defined as the ability to replicate measure- ments consistently, which is crucial for developing reliable tests and analyses. Inconsistent outcomes undermine the effectiveness of any test, preventing valid comparisons between data sets. While reliability ensures consistency, validity assesses the accuracy of measure- ments, and the two concepts are distinct. A test may be reliable, consistently producing results, but still lack validity if those results are not accurate. In industrial engineering, reliability is paramount. It denotes a system’s ability to perform its intended task effectively over time, with minimal failure. Given that modern products often consist of multiple interconnected components, the risk of failure increases if any individual part malfunctions. Therefore, a product is deemed reliable when it con- sistently performs its functions throughout its expected lifespan. The core of reliability theory is built upon the principles of measuring, analyzing, and evaluating the perfor- mance of products and systems, ensuring that they meet both operational and longevity standards. In fields such as probability, statistics, economics, survival analysis, and reliability theory, key concepts such as symmetry, asymmetry, and stochastic comparison of proba- bility distributions are integral to the analysis. Symmetry and asymmetry are particularly significant in reliability analysis, where asymmetric data present greater challenges for pre- diction compared to symmetric data. Asymmetric distributions, often seen in real-world reliability data, can complicate model building but also provide opportunities to identify outliers or anomalies that may indicate potential system failures. On the other hand, sym- metric distributions, with their more predictable patterns, can enhance the accuracy and M. S. Eliwa et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6376 3 of 24 efficiency of predictive models. Understanding the role of both symmetry and asymmetry is essential for developing effective strategies to predict and improve reliability, particu- larly in complex systems with multiple interacting components. Among the numerous life distribution types studied, the exponential distribution stands out, as an example, EL- Sagheer et al. [1], Navarro and Pellerey [2], Ghosh and Mitra [3], Bakr and Al-Babtain [4], Alqifar et al. [5], Bryson and Siddiqui [6], Barlow and Proschan [7], El-Morshedy et al. [8], Gadallah et al. [9], Mansour [10], Klefsjo [11], Kumazawa [12], Abu-Youssef et al. [13], Mahmoud and Mansour [14], Bakr [15] and Qaid et al. [16]. This distribution finds applications in various fields: estimating distances between DNA mutations (Duan et al. [17]), predicting radioactive particle decay (Poston [18]), determining molecule heights in a gas under specific conditions (Beckers et al. [19]), modeling rainfall and river flow volumes (Tomy et al. [20]). Elbatal [21] introduced a class of life distribution termed exponential better than used (EBU) and its counterpart class exponential worse than used (EWU), exploring their relationships with different life distribution groups. Elbatal investigated closure qualities under the shock model, moment inequality, and reliability operations within these classes. Some of the fundamental definitions that made it easier to derive our class include the following: Definition (1): A random variable X is said to be (i) Exponential better than used, denoted by X ∈ EBU , if F (x+ t) ≤ F (t) e −x µ , x, t > 0. (ii) Exponential better than used in increasing convex order, denoted by X ∈ EBUC, if∫ ∞ u F (x+ t) dx ≤ µe −x µ F (t) , x, t > 0, or ∫ ∞ x+t F (u) dx ≤ µe −x µ F (t) , and this leads to µWF (x+ t) ≤ µe −x µ F (t) , such that WF (x+ t) = 1 µ ∫∞ x+t F (u)du WF (x+ t) ≤ e −x µ F (t) . (iii) Exponential better than used in increasing convex in Laplace transform order, de- noted by X ∈ EBUCL, if∫ ∞ 0 e−sxWF (x+ t) dx ≤ (≥)F (t) ∫ ∞ 0 e−sxe −x µ dx, x, t > 0, s ≥ 0, or ∫ ∞ 0 e−sxWF (x+ t) dx ≤ (≥) µ µs+ 1 F (t) , x, t > 0, s ≥ 0. M. S. Eliwa et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6376 4 of 24 Remark 1. It is clear that NBU ⊂ NBUE ⊂ HNBUE. ∩ ∪ EBU ⊂ EBUC ⊂ EBUCL. Where: New better than used (NBU), New better than used in expectation (NBUE), and Harmonic new better than used in expectation (HNBUE). 1.1. Motivation and relevance The primary aim of this study is to address the current limitations in the efficiency and power of nonparametric tests for life distributions. Existing methods often fall short in terms of both test power and efficiency, motivating the development of a novel class of life distributions that improves upon these aspects. Specifically, we introduce a new class that enhances both test effectiveness and power. In contrast to previous work, where distinct classes and exponential tests have been explored (e.g., the NBRULC class in reference [8] with Moment Inequality-based testing and the EBU class in reference [21] with U- statistic-based testing), the proposed class offers a more robust framework for assessing exponentiality. This distinction highlights the novelty and potential of our approach for advancing nonparametric testing. 1.2. Outline of the paper This paper begins by highlighting the limitations of existing nonparametric tests for life distributions, specifically in terms of test efficiency and power. The primary aim is to introduce a novel class of life distributions, the EBUCL class, and develop a U- statistic-based test for exponentiality against this class. The theoretical contribution is the establishment of a framework that links Laplace transform ordering with ageing con- cepts in reliability theory. The methodological innovation includes the derivation of a U-statistic test that exhibits desirable properties such as unbiasedness and asymptotic normality, with a derived asymptotic null distribution for both complete and censored data. The computational study presents Monte Carlo simulations, which demonstrate the superior empirical power of the proposed test compared to existing methods. Additionally, Pitman’s asymptotic efficiency is computed and compared with other tests to highlight its efficiency and sensitivity to alternative distributions. The paper then applies the proposed test to real-world datasets from engineering, biostatistics, and actuarial science, showcas- ing its practical value. Finally, the test’s extension to right-censored data broadens its utility, with an emphasis on the handling of incomplete data, a common occurrence in reliability studies. 2. Assessing Alternatives to EBUCL through Testing In contrast to the associated hypothesis H1 : F suggesting non-exponential behavior but EBUCL, this section investigates the potential scenario where H0 : F represents exponentiality. The theorem presented herein serves as the basis for deriving the test statistic. M. S. Eliwa et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6376 5 of 24 Theorem 1. Suppose there exists an EBUCL random variable X with a distribution function F , thus µ2 µs+ 1 ≥ 1 2s µ(2) − 1 s3 ϕ(s)− 1 s2 µ+ 1 s3 , s ≥ 0. (1) where ϕ(s) = Ee−sX = ∞∫ 0 e−sxdF (x). Proof. Since F is EBUCL then∫ ∞ 0 e−sxWF (x+ t) dx ≤ µ µs+ 1 F (t) , x, t ≥ 0. Upon integrating both sides across the interval [0,∞) with respect to t, the result is∫ ∞ 0 ∫ ∞ 0 e−sxWF (x+ t) dxdt ≤ µ2 µs+ 1 . (2) Setting I = ∫ ∞ 0 ∫ ∞ 0 e−sxWF (x+ t) dxdt = E ∫ ∞ 0 ∫ ∞ 0 e−sx(X − x− t)I(X > x+ t)I(X > t)dxdt = E ∫ X 0 ∫ X−t 0 e−sx(X − x− t)dxdt = E ∫ X 0 ∫ X−t 0 [ Xe−sx − xe−sx − te−sx ] dxdt = E ∫ X 0 [ 1 s X + 1 s2 e−sXest − 1 s2 − 1 s t ] dt, therefore, I = 1 2s µ(2) − 1 s3 ϕ(s)− 1 s2 µ+ 1 s3 . (3) Substituting (3) into (2), we get µ2 µs+ 1 ≥ 1 2s µ(2) − 1 s3 ϕ(s)− 1 s2 µ+ 1 s3 . This completes the proof. Allow the measure of deviation from exponentiality to be suggested as follows: By setting δ(s) = 1 s2 µϕ(s) + 1 s3 ϕ(s)− 1 2 µµ(2) − 1 2s µ(2) + ( 1 s + 1)µ2 − 1 s3 . (4) M. S. Eliwa et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6376 6 of 24 Note that under H0, δ(s) = 0, while under H1, δ(s) > 0. Let X1, X2, ..., Xn be a random sample from a distribution F , the empirical estimate δ̂(s) of δ(s) can be obtained as δ̂(s) = 1 n2 n∑ i=1 n∑ j=1 { 1 s2 Xie −sXj + 1 s3 e−sXi − 1 2 XiX 2 j − 1 2s X2 i + ( 1 s + 1 ) XiXj − 1 s3 } . Based on Maclaurin series ( e−X = 1−X + X2 2! − X3 3! + ........+ (−1)n Xn n! + .. ) , we get 1 s2 Xie −sXj + 1 s3 e−sXi − 1 2 XiX 2 j − 1 2s X2 i + ( 1 s + 1 ) XiXj − 1 s3 = X1X2− 1 2 X1X 2 2 − 1 2s X2 1 . To make the test invariant, let ∆(s) = δ(s) µ3 which estimated by ∆̂(s) = δ̂(s) X 3 where X is the sample mean. Then ∆̂(s) = 1 n2X 3 n∑ i=1 n∑ j=1 { 1 s2 Xie −sXj + 1 s3 e−sXi − 1 2 XiX 2 j − 1 2s X2 i + ( 1 s + 1 ) XiXj − 1 s3 } . (5) One can note that ∆̂ (s) is an unbiased estimator of δ(s). It is easy to show that: E(∆̂(s)) = ∆(s). Now, set ϕ (Xi, Xj) = 1 s2 Xie −sXj + 1 s3 e−sXi − 1 2 XiX 2 j − 1 2s X2 i + ( 1 s + 1 ) XiXj − 1 s3 , (6) and define the symmetric kernel ψs(Xi, Xj) = 1 2! ∑ ϕs (Xi, Xj) , where the Un-statistic provided by is equal to ∆̂(s) in (5) where the summation over all arrangements of Xi, Xj Un = 1 (n2 ) n∑ i 0 using the randomly right censored data ϕ̂c = 1 s2 µϕ(s) + 1 s3 ϕ(s)− 1 2 µµ(2) − 1 2s µ(2) + ( 1 s + 1 ) µ2 − 1 s3 , where ϕ(s) = ∞∫ 0 e−sxdFn(x). To facilitate computation, ϕ̂c could be reformulated as ϕ̂c = 1 s2 Ωη + 1 s3 η − 1 2 ΩΦ− 1 2s Φ+ ( 1 s + 1)Ω2 − 1 s3 , where Ω = n∑ k=1 [ k−1∏ m=1 Cδ(m) m ( Z(k) − Z(k−1) ) ], Φ = 2 n∑ i=1 [ i−1∏ v=1 Z(i)C δ(v) v ( Z(i) − Z(i−1) ) ], η = n∑ j=1 e−sZ(j) [ j−2∏ p=1 Cδ(p) p − j−1∏ p=1 Cδ(p) p ], and dFn (Zj) = Fn(Zj−1)− Fn(Zj), ck = [n− k] [n− k + 1]−1 . To make the test invarient, let ∆̂c = ϕ̂c Z̄3 , where Z = n∑ i=1 Z(i) n . (12) Table 4 and Figure 2 exhibit the critical percentiles of the ∆̂c test corresponding to sample sizes of n = 10(10)100. By utilizing the standard exponential distribution and performing 10,000 replications using the Mathematica 12 program, critical values for the null Monte Carlo distribution were derived at s = 0.97. As depicted in Figure 2 and detailed in Table 4, the critical values exhibit an upward trend with higher confidence levels and a declining pattern as the sample size increases. M. S. Eliwa et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6376 13 of 24 Table 4. The upper percentile of ∆̂c with 10000 replications at s = 0.97. n 90% 95% 99% 10 3.24668 7.38013 24.6683 20 1.58340 3.25286 8.38960 30 1.16386 2.38361 5.82690 40 0.88624 1.81084 4.04146 50 0.68573 1.43970 3.43635 60 0.61884 1.30761 3.00585 70 0.52197 1.11054 2.62166 80 0.48741 0.97166 2.21053 90 0.42521 0.90854 1.96633 100 0.39507 0.85163 1.90046 Table 4 illustrates how the critical values rise with increasing confidence levels and fall with larger sample numbers. 20 40 60 80 100 0 5 1 0 1 5 2 0 2 5 n C .V 20 40 60 80 100 0 5 1 0 1 5 2 0 2 5 n C .V 20 40 60 80 100 0 5 1 0 1 5 2 0 2 5 n C .V 90% 95% 99% Figure 2. Sample size, confidence levels, and critical values in relation to each other under censored data. As demonstrated in Table 4 and Figure 2, there is a noticeable trend where the critical values escalate with higher confidence levels and diminish with greater sample sizes. 5.1. Estimates of test power ∆c(s) By examining the parameter values of θ at n = 10, 20, and 30 across three distinct distributions, say Weibull, LFR, and gamma, using 10,000 samples, the effectiveness of our test was assessed at a significance threshold of α = 0.05. The results displayed in M. S. Eliwa et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6376 14 of 24 Table 5 indicate that our test, ∆c(0.97)’s, exhibited commendable power estimates for all alternative scenarios. Table 5. Power estimates of ∆c(0.97). n θ Weibull LFR Gamma 10 2 3 4 0.9832 0.9999 1.0000 0.9975 0.9995 1.0000 0.9934 0.9980 0.9999 20 2 3 4 0.9710 0.9994 1.0000 0.9913 0.9990 0.9997 0.9836 0.9945 0.9986 30 2 3 4 0.9698 0.9988 1.0000 0.9897 0.9978 0.9996 0.9792 0.9923 0.9981 6. Applications of Sustainability Data from Censored and Uncensored Observations to Real-World Scenarios Statistical analysis of censored data can pose challenges due to missing or inadequate data, often necessitating special methods and assumptions. In contrast, unsupervised (or unfiltered) data alleviates these challenges, as it is devoid of such issues, thus streamlining the analysis process. 6.1. Uncensored data 6.1.1. Dataset I: Electrical data The experimental failure periods, expressed in seconds, for two distinct kinds of electrical insulation subjected to continuous voltage stress, were taken into consideration by Alsadat et al. [26]. Each type of electrical insulation has thirty samples, all of which were examined and documented. • The initial failure rates for type X are as follows: 0.097, 0.014, 0.03, 0.134, 0.240, 0.084, 0.146, 0.024, 0.045, 0.004, 0.099, 0.277, 0.472, 0.094, 0.023, 0.146, 0.030, 0.031, 0.104, 0.105, 0.036, 0.065, 0.022, 0.098, 0.178, 0.059, 0.014, 0.007, 0.007, 0.286. • The second type Y ’s failure rates can be listed as: 0.084, 0.236, 0.315, 0.199, 0.252, 0.103, 0.455, 0.135, 0.348, 0.321, 0.166, 0.04, 0.027, 0.519, 0.017, 0.821, 0.942, 0.27, 0.008, 0.03, 0.177, 0.268, 0.18, 0.796, 0.245, 0.703, 0.045, 0.314, 0.281, 0.652. Non-parametric plots of types X and Y are shown in Figures 3 and 4, respectively. The behavior of the tested data is discussed and evaluated using these charts. It was observed that both situations exhibit noticeable outlier observations, which causes the data to be M. S. Eliwa et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6376 15 of 24 skewed to the right. The shape of the hazard shown in datasets X and Y showed a pattern suggestive of a trend shaped like a bathtub-unimodal failure. Histogram Plot w F re q u e n c y 0.0 0.1 0.2 0.3 0.4 0 2 4 6 8 1 0 0.0 0.1 0.2 0.3 0.4 0 2 4 6 8 1 0 −0.1 0.1 0.3 0.5 0 1 2 3 4 5 Kernel Density N = 30 Bandwidth = 0.03444 D e n s it y 0 .0 0 .2 0 .4 1 Violin Plots 0.0 0.2 0.4 0.6 0.8 1.0 0 .0 0 .4 0 .8 i/n T (i /n ) 0.0 0.2 0.4 0.6 0.8 1.0 0 .0 0 .4 0 .8 T T T Plot 0.0 0.1 0.2 0.3 0.4 Box Plot w −2 −1 0 1 2 0 .0 0 .2 0 .4 Normal Q−Q Plot Theoretical Quantiles S a m p le Q u a n ti le s Figure 3. Non-parametric plots for dataset I (type X). Histogram Plot w F re q u e n c y 0.0 0.2 0.4 0.6 0.8 0 .0 1 .0 2 .0 0.0 0.2 0.4 0.6 0.8 0 .0 1 .0 2 .0 −0.2 0.2 0.6 1.0 0 .0 0 .5 1 .0 1 .5 2 .0 Kernel Density N = 30 Bandwidth = 0.07833 D e n s it y 0 .0 0 .4 0 .8 1 Violin Plots 0.0 0.2 0.4 0.6 0.8 1.0 0 .0 0 .4 0 .8 i/n T (i /n ) 0.0 0.2 0.4 0.6 0.8 1.0 0 .0 0 .4 0 .8 T T T Plot 0.0 0.2 0.4 0.6 0.8 Box Plot w −2 −1 0 1 2 0 .0 0 .4 0 .8 Normal Q−Q Plot Theoretical Quantiles S a m p le Q u a n ti le s Figure 4. Non-parametric plots for dataset I (type Y ). For data type X, the computed value ∆̂ = 9.03302 significantly exceeds the critical thresh- old indicated in Table 2. At the α = 0.05 significance level, the data validates the validity of the EBUCL feature. Additionally, for data type Y , ∆̂ = 2.66437 exceeds the cru- cial threshold shown in Table 2. This implies that rather than exhibiting exponential M. S. Eliwa et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6376 16 of 24 development as claimed in H1, the data collection exhibits an EBUCL feature, which we subsequently accept. 6.1.2. Dataset II: Variations in heights between plants Investigate the extensively documented Darwin dataset (Fisher [27]), depicting height discrepancies among plants from the same pair cultivated in a shared pot versus those seeded separately. The values in the dataset include 4.9, −6.7, 0.8, 1.6, 2.3, 2.8, 4.1, 1.4, 2.9, 0.6, 5.6, 2.4, 7.5, 6.0, and −4.8. Non-parametric plots were generated in Figure 5 to explore the data’s characteristics. It is observed that the data contains outliers, resulting in an asymmetric distribution with a bimodal shape. Additionally, TTT plots reveal a bathtub-shaped hazard. Histogram Plot w F re q u e n c y −6 −2 0 2 4 6 8 0 .0 0 0 .0 5 0 .1 0 0 .1 5 −6 −2 0 2 4 6 8 0 .0 0 0 .0 5 0 .1 0 0 .1 5 −10 −5 0 5 10 0 .0 0 0 .0 6 0 .1 2 Kernel Density N = 15 Bandwidth = 1.329 D e n s it y − 6 − 2 2 6 1 Violin Plots 0.0 0.2 0.4 0.6 0.8 1.0 0 .0 0 .4 0 .8 i/n T (i /n ) 0.0 0.2 0.4 0.6 0.8 1.0 0 .0 0 .4 0 .8 T T T Plot −6 −2 0 2 4 6 8 Box Plot w −1 0 1 − 6 − 2 2 6 Normal Q−Q Plot Theoretical Quantiles S a m p le Q u a n ti le s Figure 5. Non-parametric plots for data set II. In this context, the calculated value, ∆̂ = 16.4593, notably surpasses the critical value outlined in Table 2. Such data aligns with the EBUCL characteristic, therefore confirming its validity at the α = 0.05 significance level. 6.1.3. Dataset III: Strength of single carbon fibers This section examined two datasets employed by Kundu and Gupta [28] and presented in Badar and Priest [29]. Set A entails the assessment of single carbon fibers’ strength under tension, utilizing gauge lengths of 20 mm, measured in GPA. Set B, on the other hand, involves the evaluation of single carbon fibers’ strength, expressed in GPA, after stress testing at gauge lengths of 10 mm. The values in the dataset A include: 1.312, M. S. Eliwa et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6376 17 of 24 1.314, 1.479, 1.552, 1.700, 1.803, 1.861, 1.865, 1.944, 1.958, 1.966, 1.997, 2.006, 2.021, 2.027, 2.055, 2.063, 2.098, 2.140, 2.179, 2.224, 2.240, 2.253, 2.270, 2.272, 2.274, 2.301, 2.301, 2.359, 2.382, 2.382, 2.426, 2.434, 2.435, 2.478, 2.490, 2.511, 2.514, 2.535, 2.554, 2.566, 2.570, 2.586, 2.629, 2.633, 2.642, 2.648, 2.684, 2.697, 2.726, 2.770, 2.773, 2.800, 2.809, 2.818, 2.821, 2.848, 2.880, 2.954, 3.012, 3.067, 3.084, 3.090. Where are the values in the dataset B include: 1.901, 2.132, 2.203, 2.228, 2.257, 2.350, 2.361, 2.396, 2.397, 2.445, 2.454, 2.474, 2.518, 2.522, 2.525, 2.532, 2.575, 2.614, 2.616, 2.618, 2.624, 2.659, 2.675, 2.738, 2.740, 2.856, 2.917, 2.928, 2.937, 2.937, 2.977, 2.996, 3.030, 3.125, 3.139, 3.145, 3.220, 3.223, 3.235, 3.243, 3.264, 3.272, 3.294, 3.332, 3.346, 3.377, 3.408, 3.435, 3.493, 3.501, 3.537, 3.554, 3.562, 3.628, 3.852, 3.871, 3.886, 3.971, 4.024, 4.027, 4.225, 4.395, 5.020. Figures 6 and 7 were utilized to create non-parametric plots for an in-depth examination of the data’s features. Notably, dataset B displays outliers, in contrast to dataset A. Moreover, TTT plots illustrate a hazard with an increasing shape. Histogram Plot w F re q u e n c y 1.5 2.0 2.5 3.0 0 .0 0 .4 0 .8 1.5 2.0 2.5 3.0 0 .0 0 .4 0 .8 1.0 2.0 3.0 0 .0 0 .4 0 .8 Kernel Density N = 63 Bandwidth = 0.167 D e n s it y 1 .5 2 .0 2 .5 3 .0 1 Violin Plots 0.0 0.2 0.4 0.6 0.8 1.0 0 .0 0 .4 0 .8 i/n T (i /n ) 0.0 0.2 0.4 0.6 0.8 1.0 0 .0 0 .4 0 .8 T T T Plot 1.5 2.0 2.5 3.0 Box Plot w −2 −1 0 1 2 1 .5 2 .0 2 .5 3 .0 Normal Q−Q Plot Theoretical Quantiles S a m p le Q u a n ti le s Figure 6. Non-parametric plots for data set III-A. M. S. Eliwa et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6376 18 of 24 Histogram Plot w F re q u e n c y 2.0 3.0 4.0 5.0 0 .0 0 .2 0 .4 0 .6 0 .8 2.0 3.0 4.0 5.0 0 .0 0 .2 0 .4 0 .6 0 .8 1 2 3 4 5 0 .0 0 .2 0 .4 Kernel Density N = 63 Bandwidth = 0.244 D e n s it y 2 .0 3 .0 4 .0 5 .0 1 Violin Plots 0.0 0.2 0.4 0.6 0.8 1.0 0 .0 0 .4 0 .8 i/n T (i /n ) 0.0 0.2 0.4 0.6 0.8 1.0 0 .0 0 .4 0 .8 T T T Plot 2.0 3.0 4.0 5.0 Box Plot w −2 −1 0 1 2 2 .0 3 .0 4 .0 5 .0 Normal Q−Q Plot Theoretical Quantiles S a m p le Q u a n ti le s Figure 7. Non-parametric plots for data set III-B. The observed value of ∆̂ = 0.0649338 falls below the critical value specified in Table 2 for dataset A. This implies the presence of noticeable exponential patterns in the data. In contrast, the observed value of ∆̂ = −0.0608629 for dataset B deviates from the critical value in Table 2, further highlighting the discernible presence of exponential patterns in the data. 6.1.4. Dataset IV: Tensile strength of fiberglass The dataset presents outcomes from tests performed at the National Physical Laboratory in England, focusing on the tensile strengths of 1.5 cm glass fibers. This data collection has been addressed by Adepoju et al. [30]. The data can be listed as: 0.55, 0.93, 1.25, 1.36, 1.49, 1.52, 1.58, 1.61, 1.64, 1.68, 1.73, 1.81, 1.04, 1.27, 1.39, 1.49, 1.53, 1.59, 1.61, 1.66, 1.68, 1.76, 1.82, 2.01, 0.77, 1.11, 1.28, 1.42, 1.50, 1.54, 1.60, 1.62, 1.66, 1.69, 1.76, 1.84, 2.24, 0.81, 1.13, 1.29, 1.48, 1.50, 1.55, 1.61, 1.62, 1.66, 1.70, 1.77, 1.84, 0.84, 1.24, 1.30, 1.48, 1.51, 1.55, 1.61, 1.63, 1.67, 1.70, 1.78, 1.89, 0.74, 2.00. Figure 8 is employed to generate non-parametric plots for a thorough exploration of the data’s characteristics. Noteworthy is the presence of extreme in dataset. Additionally, the TTT plots depict a hazard exhibiting an increasing shape. M. S. Eliwa et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6376 19 of 24 Histogram Plot w F re q u e n c y 0.5 1.0 1.5 2.0 0 .0 0 .5 1 .0 1 .5 2 .0 0.5 1.0 1.5 2.0 0 .0 0 .5 1 .0 1 .5 2 .0 0.5 1.0 1.5 2.0 2.5 0 .0 0 .5 1 .0 1 .5 Kernel Density N = 63 Bandwidth = 0.09091 D e n s it y 0 .5 1 .0 1 .5 2 .0 1 Violin Plots 0.0 0.2 0.4 0.6 0.8 1.0 0 .0 0 .4 0 .8 i/n T (i /n ) 0.0 0.2 0.4 0.6 0.8 1.0 0 .0 0 .4 0 .8 T T T Plot 0.5 1.0 1.5 2.0 Box Plot w −2 −1 0 1 2 0 .5 1 .0 1 .5 2 .0 Normal Q−Q Plot Theoretical Quantiles S a m p le Q u a n ti le s Figure 8. Non-parametric plots for data set IV. The critical value obtained from Table 2, which stands at ∆̂ = 0.339959, exceeds our determined value. Therefore, the evident exponential nature of the data becomes apparent. 6.1.5. Dataset V: Milk produced by SINDI cows This dataset comprises the cumulative milk production from 107 SINDI race cows since their initial birth. The details can be referenced in Cordeiro and Birto [31]. Specifically, the dataset is as follows: 0.4365, 0.4260, 0.5140, 0.6907, 0.7471, 0.2605, 0.6196, 0.8781, 0.4990, 0.6058, 0.6891, 0.5770, 0.5394, 0.1479, 0.2356, 0.6012, 0.1525, 0.5483, 0.6927, 0.7261, 0.3323, 0.0671, 0.2361, 0.4800, 0.5707, 0.7131, 0.5853, 0.6768, 0.5350, 0.4151, 0.6789, 0.4576, 0.3259, 0.2303, 0.7687, 0.4371, 0.3383, 0.6114, 0.3480, 0.4564, 0.7804, 0.3406, 0.4823, 0.5912, 0.5744, 0.5481, 0.1131, 0.7290, 0.0168, 0.5529, 0.4530, 0.3891, 0.4752, 0.3134, 0.3175, 0.1167, 0.6750, 0.5113, 0.5447, 0.4143, 0.5627, 0.5150, 0.0776, 0.3945, 0.4553, 0.4470, 0.5285, 0.5232, 0.6465, 0.0650, 0.8492, 0.8147, 0.3627, 0.3906, 0.4438, 0.4612, 0.3188, 0.2160, 0.6707, 0.6220, 0.5629, 0.4675, 0.6844, 0.3413, 0.4332, 0.0854, 0.3821, 0.4694, 0.3635, 0.4111, 0.5349, 0.3751, 0.1546, 0.4517, 0.2681, 0.4049, 0.5553, 0.5878, 0.4741, 0.3598, 0.7629, 0.5941, 0.6174, 0.6860, 0.0609, 0.6488, 0.2747. Figure 9 is utilized to create non-parametric plots for a comprehensive analysis of the data’s characteristics. Notably, there are some extremes observed in the dataset. Furthermore, M. S. Eliwa et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6376 20 of 24 the TTT plots illustrate a hazard with an increasing shape. Histogram Plot w F re q u e n c y 0.5 1.0 1.5 2.0 0 .0 0 .5 1 .0 1 .5 2 .0 0.5 1.0 1.5 2.0 0 .0 0 .5 1 .0 1 .5 2 .0 0.5 1.0 1.5 2.0 2.5 0 .0 0 .5 1 .0 1 .5 Kernel Density N = 63 Bandwidth = 0.09091 D e n s it y 0 .5 1 .0 1 .5 2 .0 1 Violin Plots 0.0 0.2 0.4 0.6 0.8 1.0 0 .0 0 .4 0 .8 i/n T (i /n ) 0.0 0.2 0.4 0.6 0.8 1.0 0 .0 0 .4 0 .8 T T T Plot 0.5 1.0 1.5 2.0 Box Plot w −2 −1 0 1 2 0 .5 1 .0 1 .5 2 .0 Normal Q−Q Plot Theoretical Quantiles S a m p le Q u a n ti le s Figure 9. Non-parametric plots for data set V. The critical value depicted in Table 2 was found to be exceeded by ∆̂ = 1.82338. Accordingly, our agreement aligns with hypothesis H1, suggesting that the data collection is characterized by EBUCL rather than exponential growth. 6.2. Censored data 6.2.1. Dataset VI: Incurable lung cancer As reported by Lagakos and Williams ([32]) and Lee and Wolfe ([33]), out of 61 patients treated with cyclophosphamide for incurable lung cancer, the following statistics were observed: 33 patients had unfiltered observations, while 28 patients had their treatment stopped due to deteriorating health, leading to censored observations. Observational cen- sorship: 0.14, 0.14, 0.29, 0.43, 0.57, 0.57, 1.86, 3.00, 3.00, 3.29, 3.29, 6.00, 6.00, 6.14, 8.71, 10.57, 11.86, 15.57, 16.57, 17.29, 18.71, 21.29, 23.86, 26.00, 27.57, 32.14, 33.14, 47.29. Un- censored observations: 0.43, 2.86, 3.14, 3.14, 3.43, 3.43, 3.71, 3.86, 6.14, 6.86, 9.00, 9.43, 10.71, 10.86, 11.14, 13.00, 14.43, 15.71, 18.43, 18.57, 20.71, 29.14, 29.71, 40.57, 48.57, 49.43, 53.86, 61.86, 66.57, 68.71, 68.96, 72.86, 72.86. When considering all survival data, includ- ing both censored and uncensored observations, our finding, ∆c(0.97) = −2.67235× 1049, falls below the critical threshold indicated in Table 4. Hence, the exponential pattern within the data becomes evident. M. S. Eliwa et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6376 21 of 24 6.2.2. Dataset VII: Tongue cancer patients The dataset presents the estimated time of death (in weeks) for tongue cancer patients with aneuploidy DNA profiles. Previous references utilizing this data include Sickle-Santanello et al. [34] and Klein and Moeschaberger [35]. The dataset contains 51 observations: 1, 3, 3, 4, 10, 13, 16, 16, 24, 28, 30, 30, 32, 41, 51, 61+, 65, 67, 72, 73, 74+, 77, 79+, 80+, 81+, 87+, 87+, 89+, 91, 93+, 96, 97+, 100, 101+, 104, 104+, 109+, 120+, 150+, 131+, 157, 167, 13, 231+, 240+, 24, 27, 70, 88+, 108+, 400. The result of ∆c(0.97) = −3.35085×1020 is derived, falling below the critical threshold detailed in Table 4. Hence, this supports the null hypothesis regarding the exponential characteristic, leading to the rejection of the alternative hypothesis of EBUCL. 7. Conclusion This study provided a comprehensive examination of the exponential distribution, a fundamental concept in statistical theory with wide applications in reliability theory, life testing, and stochastic processes. The research focused on the development of innovative nonparametric testing techniques to assess whether data conform to the properties of the exponential distribution and the EBUCL class. Both complete and censored datasets were thoroughly analyzed, and the study demonstrated the asymptotic normality of the pro- posed test. Additionally, the paper calculated upper percentile values for the test statistics using Monte Carlo simulations, and evaluated the power of the test against alternative distributions, including the LFR, Gamma, and Weibull distributions. The results were further validated through the calculation of Pitman’s asymptotic relative efficiencies and applied to real-world datasets, confirming the test’s practical utility and reliability in var- ious fields. The findings have important implications for reliability engineering, where the exponential distribution is widely used to model constant failure rates, allowing engineers to effectively schedule maintenance and predict potential system failures. This study also highlighted the relevance of exponentiality testing in healthcare, where it plays a crucial role in understanding disease trajectories, evaluating interventions, and forecasting public health outcomes such as epidemic frequencies and vaccine efficacy. By providing a rigorous statistical framework for testing exponentiality and examining the reliability of systems in both engineering and healthcare, this research contributes valuable insights that can guide decision-making, enhance predictive models, and foster more effective maintenance and healthcare planning strategies. The study emphasizes the importance of exponen- tiality testing in real-world applications and lays the groundwork for future research that could expand these methodologies across other sectors of applied mathematics and relia- bility theory. Acknowledgements This project was supported by the deanship of scientific research at Prince Sattam bin Abdulaziz University, Al-Kharj, Saudi Arabia. This study is supported via funding from M. S. Eliwa et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6376 22 of 24 Prince Sattam bin Abdulaziz University project number (PSAU/2025/R/1446). Data Availability: Data are contained within the article. 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