Integrating XGBoost with the new cotangent-$ \Phi $ generated family of distributions: a unified frequentist-Bayesian approach with applications

Aijaz Ahmad; Farouq Mohammad A. Alam; Andaç Batur Çolak; Aafaq A. Rather; Mahmoud E. Bakr; Aijaz Ahmad; Farouq Mohammad A. Alam; Andaç Batur Çolak; Aafaq A. Rather; Mahmoud E. Bakr

doi:10.3934/math.2026921

AIMS Mathematics

2026, Volume 11, Issue 7: 22845-22896. doi: 10.3934/math.2026921

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Integrating XGBoost with the new cotangent-$ \Phi $ generated family of distributions: a unified frequentist-Bayesian approach with applications

1.
Lincoln University College, Petaling Jaya-47301, Selangor, Malaysia
2.
Department of Statistics, Faculty of Science, King Abdulaziz University, Saudi Arabia
3.
Department of Information Systems and Technologies, Faculty of Computer and Information Sciences, Niğde Ömer Halisdemir University, Niğde 51240, Türkiye
4.
Symbiosis Statistical Institute, Symbiosis International (Deemed University), Pune, India
5.
Department of Statistics and Operations Research, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

Received: 25 April 2026 Revised: 07 July 2026 Accepted: 09 July 2026 Published: 28 July 2026
MSC : 62A86, 62D05, 62F12, 62F25, 62G07, 62L05, 62L10, 62L12

This study introduces the cotangent-$ \Phi $ (NC-$ \Phi $) distribution, which establishes a new type of probability distribution for statistical modeling complex real-world data. The proposed method uses trigonometric transformations to systematically manipulate the skewness and tail characteristics of traditional distributions, thus overcoming the difficulties that often arise in existing models. A complete mathematical theory is formulated, with series expansions of the cumulative distribution function and the probability density function. Based on the Weibull distribution, a new cotangent-Weibull distribution (NCWD) is introduced, and its major statistical features are investigated. A machine learning layer was developed to complement the analytical properties of the formulated NCWD. Parameter estimation is carried out in both frequentist and Bayesian approaches, and a Monte Carlo simulation study is conducted to evaluate and compare the performance of the estimators under different sample sizes. The validity of the proposed model is established through applications to three different real-world datasets: the active repair times (in hours) of an aerial communication transceiver, the inter-arrival times between vehicles passing a fixed point on a roadway, and the susceptibility index data of peppermint packages irradiated by gamma and microwave radiation under non-choice packaging conditions. In each scenario, the NCWD outperforms several other competing models in terms of the goodness-of-fit, thus establishing its appropriateness for reliability, transportation, and irradiation data analyses.
- cotangent function,
- Weibull distribution,
- moments,
- Bayesian analysis,
- XGBoost,
- data analysis
Citation: Aijaz Ahmad, Farouq Mohammad A. Alam, Andaç Batur Çolak, Aafaq A. Rather, Mahmoud E. Bakr. Integrating XGBoost with the new cotangent-$ \Phi $ generated family of distributions: a unified frequentist-Bayesian approach with applications[J]. AIMS Mathematics, 2026, 11(7): 22845-22896. doi: 10.3934/math.2026921

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Abstract

This study introduces the cotangent-$ \Phi $ (NC-$ \Phi $) distribution, which establishes a new type of probability distribution for statistical modeling complex real-world data. The proposed method uses trigonometric transformations to systematically manipulate the skewness and tail characteristics of traditional distributions, thus overcoming the difficulties that often arise in existing models. A complete mathematical theory is formulated, with series expansions of the cumulative distribution function and the probability density function. Based on the Weibull distribution, a new cotangent-Weibull distribution (NCWD) is introduced, and its major statistical features are investigated. A machine learning layer was developed to complement the analytical properties of the formulated NCWD. Parameter estimation is carried out in both frequentist and Bayesian approaches, and a Monte Carlo simulation study is conducted to evaluate and compare the performance of the estimators under different sample sizes. The validity of the proposed model is established through applications to three different real-world datasets: the active repair times (in hours) of an aerial communication transceiver, the inter-arrival times between vehicles passing a fixed point on a roadway, and the susceptibility index data of peppermint packages irradiated by gamma and microwave radiation under non-choice packaging conditions. In each scenario, the NCWD outperforms several other competing models in terms of the goodness-of-fit, thus establishing its appropriateness for reliability, transportation, and irradiation data analyses.

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