MDPI - Publisher of Open Access Journals

38 pages, 1794 KB

Open AccessArticle

On a New NBRUL*t₀ Reliability Class and Efficient Tests of Exponentiality: Mathematical Theory and Applications

by Mahmoud M. Ramadan, Rashad M. EL-Sagheer, Mahmoud E. Bakr, Yusra A. Tashkandy, Oluwafemi Samson Balogun and Walid B. H. Etman

Mathematics 2026, 14(9), 1469; https://doi.org/10.3390/math14091469 - 27 Apr 2026

Viewed by 402

Abstract

In this paper, we introduce a new age-dependent reliability class, termed the new better (worse) than renewal used in Laplace transform order after age

t_{0}

(

N B R U L^{*} t_{0}

). This class extends existing aging notions by [...] Read more.

In this paper, we introduce a new age-dependent reliability class, termed the new better (worse) than renewal used in Laplace transform order after age

t_{0}

(

N B R U L^{*} t_{0}

). This class extends existing aging notions by characterizing lifetime distributions through renewal-based Laplace transform ordering beyond a specified age threshold. Several theoretical properties of the proposed class are established, including its relationships with classical aging classes. A goodness-of-fit test for exponentiality against the

N B R U L^{*} t_{0}

alternative is developed using the framework of U-statistics, yielding a scale-invariant test statistic with a tractable asymptotic distribution. The asymptotic normality and Pitman asymptotic efficiency of the proposed test are derived, demonstrating superior efficiency relative to several existing nonparametric competitors. Extensive Monte Carlo simulations are conducted to obtain critical values and to assess the power performance of the test under both complete and randomly right-censored samples. The results indicate that the proposed test exhibits high power and robustness, particularly in the presence of aging effects and censoring. Applications to real engineering and medical datasets illustrate the practical relevance of the

N B R U L^{*} t_{0}

class in reliability analysis and survival studies. Full article

► Show Figures

Figure 1

33 pages, 672 KB

Open AccessArticle

A Laplace Transform-Based Test for Exponentiality Against the EBUCL Class with Applications to Censored and Uncensored Data

by Walid B. H. Etman, Mahmoud E. Bakr, Arwa M. Alshangiti, Oluwafemi Samson Balogun and Rashad M. EL-Sagheer

Mathematics 2025, 13(21), 3379; https://doi.org/10.3390/math13213379 - 23 Oct 2025

Viewed by 756

Abstract

This paper proposes a novel statistical test for evaluating exponentiality against the recently introduced EBUCL (Exponential Better than Used in Convex Laplace transform order) class of life distributions. The EBUCL class generalizes classical aging concepts and provides a flexible framework for modeling various [...] Read more.

This paper proposes a novel statistical test for evaluating exponentiality against the recently introduced EBUCL (Exponential Better than Used in Convex Laplace transform order) class of life distributions. The EBUCL class generalizes classical aging concepts and provides a flexible framework for modeling various non-exponential aging behaviors. The test is constructed using Laplace transform ordering and is shown to be effective in distinguishing exponential distributions from EBUCL alternatives. We derive the test statistic, establish its asymptotic properties, and assess its performance using Pitman’s asymptotic efficiency under standard alternatives, including Weibull, Makeham, and linear failure rate distributions. Critical values are obtained through extensive Monte Carlo simulations, and the power of the proposed test is evaluated and compared with existing methods. Furthermore, the test is extended to handle right-censored data, demonstrating its robustness and practical applicability. The effectiveness of the procedure is illustrated through several real-world datasets involving both censored and uncensored observations. The results confirm that the proposed test is a powerful and versatile tool for reliability and survival analysis. Full article

► Show Figures

Figure 1

23 pages, 3220 KB

Open AccessArticle

A Novel Statistical Test for Life Distribution Analysis: Assessing Exponentiality Against EBUCL Class with Applications in Sustainability and Reliability Data

by Walid B. H. Etman, Mahmoud E. Bakr, Oluwafemi Samson Balogun and Rashad M. EL-Sagheer

Axioms 2025, 14(2), 140; https://doi.org/10.3390/axioms14020140 - 17 Feb 2025

Cited by 1 | Viewed by 1034

Abstract

A product’s lifespan may be ascertained by analyzing the dependability and aging class of its life distribution. Sustainability metrics are also crucial for assessing the impact on the environment and creating resource-saving strategies. Researchers extensively rely on statistical testing when making judgments; nonparametric [...] Read more.

A product’s lifespan may be ascertained by analyzing the dependability and aging class of its life distribution. Sustainability metrics are also crucial for assessing the impact on the environment and creating resource-saving strategies. Researchers extensively rely on statistical testing when making judgments; nonparametric tests are particularly useful, since they may be used for a broad variety of datasets and do not require knowledge of the data distribution. To provide effective assessments for well-informed decisions, this study introduces a novel life distribution class called “Exponential Better than Used in Increasing Convex in Laplace Transform Order (EBUCL)”. In this framework, a novel test statistic utilizing the moment inequalities method is introduced to evaluate exponentiality against the EBUCL class. By analyzing its asymptotic behavior, critical values are calculated for sample sizes between 5 and 100. Pitman’s asymptotic efficiency was taken into consideration by calculating the test’s power and comparing it with other tests through the use of simulation studies employing standard reliability distributions. In conclusion, this study addresses the treatment of right-censored data and applied the suggested test approach to real datasets across many domains. Our experimental results have practical implications for sustainability data assessments in engineering and the biological sciences. Full article

► Show Figures

Figure 1

16 pages, 2449 KB

Open AccessArticle

Reliability Analysis for Unknown Age Class of Lifetime Distribution with Real Applications in Medical Science

by Mahmoud E. Bakr, Oluwafemi Samson Balogun, Asmaa A. El-Toony and Alaa. M. Gadallah

Symmetry 2024, 16(11), 1514; https://doi.org/10.3390/sym16111514 - 11 Nov 2024

Cited by 7 | Viewed by 1362

Abstract

Analyzing the reliability of the aging class of life distribution provides important information about how long a product lasts and sustainability measures that are essential for determining the environmental impact and formulating resource-saving plans. The study emphasizes the goodness-of-fit technique of the nonparametric [...] Read more.

Analyzing the reliability of the aging class of life distribution provides important information about how long a product lasts and sustainability measures that are essential for determining the environmental impact and formulating resource-saving plans. The study emphasizes the goodness-of-fit technique of the nonparametric test for the NBRU_mgf class because age data are crucial for applications. Evaluations were conducted using the test’s asymptotic properties and Pitman efficiency methodology for some selected asymmetric probability models, and the outcomes were compared with those of alternative methods. We assessed the test’s power against widely used reliability distributions for some well-known alternative asymmetric distributions, including the Weibull, Gamma, and linear failure rate (LFR) distributions, and provided percentiles for both censored and uncensored data. This study shows the efficacy of the test in various sectors using real-world datasets and comprehensive tables of test statistics. Full article

(This article belongs to the Section B: Mathematics)

► Show Figures

Figure 1

14 pages, 886 KB

Open AccessArticle

Applying the Laplace Transform Procedure, Testing Exponentiality against the NBRU_mgf Class

by Naglaa A. Hassan, Mayar M. Said, Rasha Abd El-Wahab Attwa and Taha Radwan

Mathematics 2024, 12(13), 2045; https://doi.org/10.3390/math12132045 - 30 Jun 2024

Cited by 1 | Viewed by 1393

Abstract

This paper addresses a hypothesis testing problem for comparing exponentially distributed data against a new class termed “New Better than Renewal Used in Moment Generating Function” (

N B R U_{m g f}

). A measure of departure from exponentiality is constructed [...] Read more.

This paper addresses a hypothesis testing problem for comparing exponentially distributed data against a new class termed “New Better than Renewal Used in Moment Generating Function” (

N B R U_{m g f}

). A measure of departure from exponentiality is constructed using the Laplace transform, followed by the development of a U-statistic-based test for the hypothesis. Additionally, a test based on the goodness of fit approach is examined as a special case. The asymptotic normality of the proposed statistic is introduced, and Pitman’s asymptotic efficiency of the two tests is computed and compared with other tests. Percentiles of the test statistics are computed for certain sample sizes in the case of complete data, and the powers of the tests are computed for popular reliability distributions. Finally, practical applications of the proposed tests are demonstrated in multiple cases. Full article

(This article belongs to the Special Issue Parametric and Nonparametric Statistics: From Theory to Applications)

► Show Figures

Figure 1

11 pages, 765 KB

Open AccessArticle

Statistical Modeling for Some Real Applications in Reliability Analysis Using Non-Parametric Hypothesis Testing

by Mahmoud E. Bakr

Symmetry 2023, 15(9), 1735; https://doi.org/10.3390/sym15091735 - 10 Sep 2023

Cited by 3 | Viewed by 2233

Abstract

Probability life distributions usually describe the time to an event or survival time. Therefore, these life distributions play a crucial role in the analysis and projection of maximum life expectancy using the goodness of fit approach for nonparametric hypothesis testing. This study suggests [...] Read more.

Probability life distributions usually describe the time to an event or survival time. Therefore, these life distributions play a crucial role in the analysis and projection of maximum life expectancy using the goodness of fit approach for nonparametric hypothesis testing. This study suggests a nonparametric technique to determine whether the data follow an exponential distribution or belong to the mathematical class of the moment generating function for used better than aged (UBAmgf). These tests can be applied to both censored and non-censored data. The upper percentile points of the test statistics are generated, and the suggested test’s asymptotic normality is established. Some well-known alternative asymmetric probability models are used to compute the Pitman asymptotic relative efficiency (PARE) and powers of the proposed test. To demonstrate the paper’s conclusions, some asymmetric real-world datasets are examined. Full article

► Show Figures

Figure 1

25 pages, 445 KB

Open AccessArticle

Reliability Class Testing and Hypothesis Specification: NBRULC−t_∘ Characterizations with Applications for Medical and Engineering Data Modeling

by Hana Alqifari, Mohamed S. Eliwa, Walid B. H. Etman, Mahmoud El-Morshedy, Laila A. Al-Essa and Rashad M. EL-Sagheer

Axioms 2023, 12(5), 414; https://doi.org/10.3390/axioms12050414 - 24 Apr 2023

Cited by 8 | Viewed by 2674

Abstract

Due to the complexity of the data being generated day in and day out in many practical domains, as a result of the development of scales for rating the success or failure of reliability, a new domain of reliability called the classes of [...] Read more.

Due to the complexity of the data being generated day in and day out in many practical domains, as a result of the development of scales for rating the success or failure of reliability, a new domain of reliability called the classes of life and determinant probability distributions has been presented. This article introduces novel statistical probability models for the reliability class of life test under different reliability processes in the age range

t_{\circ}

. Several probabilistic properties and features were derived and rigorously screened to test the new reliability class. According to the U-statistic, a novel hypothesis test was created to evaluate the exponentiality property. The comparative efficiency of the test according to Pitman’s asymptotic efficiency was examined and compared with other reliability classes. To prove the superiority of the new reliability class, some probability models were utilized, including the Weibull, Makeham, gamma, and linear failure rate models. Moreover, critical point simulations of the null Monte Carlo distribution and some applications of the censored and uncensored data were implemented to validate the class test listed by the reliability analysis. Full article

(This article belongs to the Special Issue Mathematical and Statistical Methods and Their Applications)

► Show Figures

Figure 1

13 pages, 446 KB

Open AccessArticle

A Moment Inequality for the NBRULC Class: Statistical Properties with Applications to Model Asymmetric Data

by Mahmoud El-Morshedy, Afrah Al-Bossly, Rashad M. EL-Sagheer, Bader Almohaimeed, Waleed B. H. Etman and Mohamed S. Eliwa

Symmetry 2022, 14(11), 2353; https://doi.org/10.3390/sym14112353 - 8 Nov 2022

Cited by 9 | Viewed by 1924

Abstract

In this paper, the moment inequalities for some aging distributions are derived based on a mathematical class entitled “a new better than renewal used in Laplace transform order in increasing convex order class (NBRULC)”. The introduced inequalities can be utilized as a [...] Read more.

In this paper, the moment inequalities for some aging distributions are derived based on a mathematical class entitled “a new better than renewal used in Laplace transform order in increasing convex order class (NBRULC)”. The introduced inequalities can be utilized as a new mathematical test for the exponentiality property versus NBRULC. If the mean life is finitely based on these inequalities, then all higher-order moments exist. Pitman’s asymptotic efficiency of the new mathematical test is derived and studied in detail for some asymmetric probability models. The new mathematical test’s power is estimated in reliability studies for a few well-known alternative asymmetric models. The problem in the case of right-censored data is also handled. After that, applying the suggested test to practical issues is demonstrated using asymmetric, real datasets. Full article

(This article belongs to the Section B: Mathematics)

► Show Figures

Figure 1

Search Results (8)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI

Saved Queries

Search Filter Reset All

Years

Feature Papers

Subjects

Journals

Article Types

Countries / Regions

Search Results (8)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI