The fractional exponential distribution: A gamma subfamily from conformable calculus

Abstract

This paper develops a fractional-calculus derivation of a constrained gamma subfamily by applying the conformable fractional derivative to the cumulative distribution function of the exponential distribution. The resulting two-parameter model, referred to as the fractional exponential (FE) distribution, has a density that is identical to a gamma density with shape parameter 2 − α and rate parameter β , where 0 < α ≤ 1 and β > 0. Therefore, the main contribution of this work is not the introduction of a wholly new distributional family, but rather a fractional derivation, interpretation, and restricted parameterization of a tractable gamma subfamily. Within this framework, we derive the cumulative distribution function, survival function, hazard function, ordinary moments, incomplete moments, residual-life moments, quantiles, mode, and Rényi entropy. Frequentist estimation is considered using maximum likelihood, method of moments, least squares, Cramér–von Mises, and Anderson–Darling criteria, while Bayesian estimation is developed using a gamma prior for β and a beta prior for α ∈ (0, 1]. A Monte Carlo simulation study compares the performance of the proposed estimators under different parameter settings and sample sizes, and bootstrap confidence intervals are reported for the frequentist methods. Finally, three real data sets are analyzed to assess the flexibility of the FE model relative to the exponential, Weibull, Lindley, and weighted exponential distributions. The results show that the FE model provides a mathematically tractable fractional interpretation of a specific gamma subfamily and offers a useful alternative perspective for lifetime data analysis.