The Ujlayan-Dixit fractional Weibull distribution: statistical properties, estimation, and applications to engineering and biomedical data

Nesreen M. Al-Olaimat; Hatim Solayman Migdadi; Afaf Alrashidi; Samirah Alzubaidi; Nesreen M. Al-Olaimat; Hatim Solayman Migdadi; Afaf Alrashidi; Samirah Alzubaidi

doi:10.3934/math.20261375

AIMS Mathematics

2026, Volume 11, Issue 10: 35429-35468. doi: 10.3934/math.20261375

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Research article

The Ujlayan-Dixit fractional Weibull distribution: statistical properties, estimation, and applications to engineering and biomedical data

1.
Department of Applied Science, Ajloun College, Al-Balqa Applied University, Ajloun 26816, Jordan
2.
Department of Mathematics, Faculty of Science, The Hashemite University, Zarqa 13133, Jordan
3.
Department of Statistics, Faculty of Science, University of Tabuk, Tabuk, Saudi Arabia
4.
Department of Mathematics, Al-Qunfudah University College, Umm Al-Qura University, Mecca, Saudi Arabia

Received: 03 August 2026 Revised: 26 September 2026 Accepted: 10 October 2026 Published: 28 October 2026
MSC : 26A33, 60E05, 62N05, 62F10

A new three-parameter Ujlayan-Dixit fractional Weibull distribution (UDFWD) is proposed by replacing the ordinary derivative in the differential equation generating the classical Weibull density with the Ujlayan-Dixit fractional (UDF) derivative of order $q \in (0, 1]$. The resulting density is expressed as a convergent infinite series involving the gamma function. Series representations are obtained for the cumulative distribution, survival, and hazard rate functions in terms of the lower and upper incomplete gamma functions, together with sufficient conditions on $(\theta, \beta, q)$ for absolute convergence. Unlike most Weibull-type extensions, whose aging behavior is examined only numerically, the proposed density is proved to be strictly log-concave for $ \theta > 1 $, $ \beta > 0 $, and $ 0 < q\leq1 $. The UDFWD therefore has a strictly increasing hazard rate and a unique mode, and the fractional parameter is shown to govern the level and the growth of the hazard through the factor $1/{q}$ and the additive constant ${(1-q)}/{q}$. Several structural properties are also derived, including the mode; the quantile function; the ordinary moments; and the Shannon, Tsallis, and Rényi entropies. The unknown parameters are estimated using the maximum likelihood, least squares, and weighted least squares methods, and a Monte Carlo simulation study provides numerical evidence that the mean square errors (MSEs) and absolute relative biases generally decrease as the sample size increases, with the maximum likelihood method attaining the smallest MSEs in most settings with moderate and large sample sizes. Finally, two real datasets, on carbon fiber tensile strength and serum creatinine, show that the UDFWD attains the smallest values of all model-selection criteria against six competing lifetime models, while remaining highly competitive on the distance-based goodness-of-fit measures.
- Ujlayan-Dixit fractional derivative,
- Weibull distribution,
- hazard rate function,
- log-concavity,
- maximum likelihood estimation,
- moments and entropy,
- Monte Carlo simulation
Citation: Nesreen M. Al-Olaimat, Hatim Solayman Migdadi, Afaf Alrashidi, Samirah Alzubaidi. The Ujlayan-Dixit fractional Weibull distribution: statistical properties, estimation, and applications to engineering and biomedical data[J]. AIMS Mathematics, 2026, 11(10): 35429-35468. doi: 10.3934/math.20261375

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Abstract

A new three-parameter Ujlayan-Dixit fractional Weibull distribution (UDFWD) is proposed by replacing the ordinary derivative in the differential equation generating the classical Weibull density with the Ujlayan-Dixit fractional (UDF) derivative of order $q \in (0, 1]$. The resulting density is expressed as a convergent infinite series involving the gamma function. Series representations are obtained for the cumulative distribution, survival, and hazard rate functions in terms of the lower and upper incomplete gamma functions, together with sufficient conditions on $(\theta, \beta, q)$ for absolute convergence. Unlike most Weibull-type extensions, whose aging behavior is examined only numerically, the proposed density is proved to be strictly log-concave for $ \theta > 1 $, $ \beta > 0 $, and $ 0 < q\leq1 $. The UDFWD therefore has a strictly increasing hazard rate and a unique mode, and the fractional parameter is shown to govern the level and the growth of the hazard through the factor $1/{q}$ and the additive constant ${(1-q)}/{q}$. Several structural properties are also derived, including the mode; the quantile function; the ordinary moments; and the Shannon, Tsallis, and Rényi entropies. The unknown parameters are estimated using the maximum likelihood, least squares, and weighted least squares methods, and a Monte Carlo simulation study provides numerical evidence that the mean square errors (MSEs) and absolute relative biases generally decrease as the sample size increases, with the maximum likelihood method attaining the smallest MSEs in most settings with moderate and large sample sizes. Finally, two real datasets, on carbon fiber tensile strength and serum creatinine, show that the UDFWD attains the smallest values of all model-selection criteria against six competing lifetime models, while remaining highly competitive on the distance-based goodness-of-fit measures.

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