The bimodal two-piece skew-normal distribution: Mathematical theory, reliability aging measures, and simulation-oriented decision analysis

Reda Elbarougy; Jondeep Das; Partha Jyoti Hazarika; Mohamed S. Eliwa; Reda Elbarougy; Jondeep Das; Partha Jyoti Hazarika; Mohamed S. Eliwa

doi:10.3934/math.2026022

AIMS Mathematics

2026, Volume 11, Issue 1: 511-542. doi: 10.3934/math.2026022

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Research article Special Issues

The bimodal two-piece skew-normal distribution: Mathematical theory, reliability aging measures, and simulation-oriented decision analysis

1.
Department of Artificial Intelligence and Data Science, College of Computer Science and Engineering, University of Ha'il, Saudi Arabia
2.
Department of Statistics, Bhattadev University, Bajali, Assam, India
3.
Department of Statistics, Dibrugarh University, Dibrugarh, Assam, India
4.
Department of Statistics and Operations Research, College of Science, Qassim University, Saudi Arabia

Received: 15 November 2025 Revised: 25 December 2025 Accepted: 29 December 2025 Published: 07 January 2026
MSC : 60E05, 62E15, 62F10, 62N05, 62P99

This study introduces a novel and adaptable bimodal class of two-piece skew-normal distributions, specifically designed to accommodate datasets with up to two modes. This paper comprehensively examines the analytical attributes of the suggested model, encompassing its cumulative distribution function, moments, moment generating function, Rényi entropy, reliability metrics with aging intensity, and other critical statistical properties. The model effectively captures asymmetric behavior, accommodating both negative and positive skewness, and is particularly well suited for leptokurtic data characterized by an increasing hazard rate. Furthermore, it adeptly manages both over- and under-dispersed data, enhancing its relevance across many domains. To improve its adaptability, extensions of the distribution concerning location and scale are also devised. Parameter estimation is conducted using the maximum likelihood approach. A detailed simulation study assesses the performance of the estimators, illustrating their asymptotic consistency and efficiency. The practical utility of the suggested distribution is demonstrated through applications to real-world datasets, where it regularly outperforms multiple existing rival models in goodness-of-fit. Finally, a likelihood ratio test is utilized to statistically validate the superiority of the presented model compared to its nested alternatives.
- statistical model,
- bimodal data modeling,
- non-symmetric and leptokurtic distributions,
- failure analysis,
- simulation,
- likelihood ratio test,
- decision-making
Citation: Reda Elbarougy, Jondeep Das, Partha Jyoti Hazarika, Mohamed S. Eliwa. The bimodal two-piece skew-normal distribution: Mathematical theory, reliability aging measures, and simulation-oriented decision analysis[J]. AIMS Mathematics, 2026, 11(1): 511-542. doi: 10.3934/math.2026022

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Abstract

This study introduces a novel and adaptable bimodal class of two-piece skew-normal distributions, specifically designed to accommodate datasets with up to two modes. This paper comprehensively examines the analytical attributes of the suggested model, encompassing its cumulative distribution function, moments, moment generating function, Rényi entropy, reliability metrics with aging intensity, and other critical statistical properties. The model effectively captures asymmetric behavior, accommodating both negative and positive skewness, and is particularly well suited for leptokurtic data characterized by an increasing hazard rate. Furthermore, it adeptly manages both over- and under-dispersed data, enhancing its relevance across many domains. To improve its adaptability, extensions of the distribution concerning location and scale are also devised. Parameter estimation is conducted using the maximum likelihood approach. A detailed simulation study assesses the performance of the estimators, illustrating their asymptotic consistency and efficiency. The practical utility of the suggested distribution is demonstrated through applications to real-world datasets, where it regularly outperforms multiple existing rival models in goodness-of-fit. Finally, a likelihood ratio test is utilized to statistically validate the superiority of the presented model compared to its nested alternatives.

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