Copula-based analysis of asymmetrically distributed joint data under a risk profile using the discrete Type-I extreme value distribution

Hend S. Shahen; Mahmoud El-Morshedy; Mohamed S. Eliwa; Mohamed F. Abouelenein; Hend S. Shahen; Mahmoud El-Morshedy; Mohamed S. Eliwa; Mohamed F. Abouelenein

doi:10.3934/era.2025203

Electronic Research Archive

2025, Volume 33, Issue 8: 4468-4494. doi: 10.3934/era.2025203

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Copula-based analysis of asymmetrically distributed joint data under a risk profile using the discrete Type-I extreme value distribution

1.
Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia
2.
Department of Statistics and Operations Research, College of Science, Qassim University, Saudi Arabia
3.
Department of Insurance and Risk Management, College of Business, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11432, Riyadh, Saudi Arabia

Received: 24 June 2025 Revised: 13 July 2025 Accepted: 21 July 2025 Published: 06 August 2025

Copulas provide a flexible framework for building bivariate probability models that reflect specific dependency patterns. This work introduces a discrete form of the Type-I extreme value (Gumbel) distribution within a copula-based structure. Key mathematical and statistical characteristics are examined, including the joint probability mass function, survival function, hazard rate, conditional expectation, joint probability generating function, and dependency properties such as positive quadrant dependence and total positivity of order two. The bivariate discrete Gumbel model demonstrates strong performance in handling asymmetric data and proves particularly useful for capturing extreme and outlier observations. Its joint hazard rate function adds further flexibility, making it suitable for modeling a range of failure rate behaviors. Parameter estimation is carried out using the maximum likelihood method, and a thorough simulation study evaluates the bias and mean squared errors across various sample sizes. To illustrate its practical relevance, the model is applied to three different real-world datasets: football match outcomes, nasal drainage severity scores, and lens defects involving surface and interior faults.
- statistical model,
- copula approach,
- extreme value distribution,
- failure analysis,
- conditional expectation,
- simulation,
- data-driven approaches,
- decision-making
Citation: Hend S. Shahen, Mahmoud El-Morshedy, Mohamed S. Eliwa, Mohamed F. Abouelenein. Copula-based analysis of asymmetrically distributed joint data under a risk profile using the discrete Type-I extreme value distribution[J]. Electronic Research Archive, 2025, 33(8): 4468-4494. doi: 10.3934/era.2025203

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Abstract

Copulas provide a flexible framework for building bivariate probability models that reflect specific dependency patterns. This work introduces a discrete form of the Type-I extreme value (Gumbel) distribution within a copula-based structure. Key mathematical and statistical characteristics are examined, including the joint probability mass function, survival function, hazard rate, conditional expectation, joint probability generating function, and dependency properties such as positive quadrant dependence and total positivity of order two. The bivariate discrete Gumbel model demonstrates strong performance in handling asymmetric data and proves particularly useful for capturing extreme and outlier observations. Its joint hazard rate function adds further flexibility, making it suitable for modeling a range of failure rate behaviors. Parameter estimation is carried out using the maximum likelihood method, and a thorough simulation study evaluates the bias and mean squared errors across various sample sizes. To illustrate its practical relevance, the model is applied to three different real-world datasets: football match outcomes, nasal drainage severity scores, and lens defects involving surface and interior faults.

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