A New One-Parameter Gamma Family Derived via Fractional Differential Equations and Its Application to Malaria Indicator Data

In this work, we use a fractional differential equation to create a new family of one-parameter probability distributions. We build a generalized version of the typical exponential-type model using methods from fractional calculus, namely, the UD fractional derivative. The suggested fractional differential equation’s closed-form solution produces a flexible probability density function (PDF) with a single scale parameter. The quantile function, moments, moment generating function (MGF), skewness and kurtosis measurements, and the cumulative distribution function (CDF) are among the fundamental statistical aspects of the new distribution that are analytically derived. Additionally, reliability metrics such as the mean residual life function, survival function, hazard rate, and reversed hazard rate are acquired. The approach of maximum likelihood is applied to parameter estimation. Applications to malaria indicator data showed the superior fitting ability of the suggested model when compared with classical distributions using goodness-of-fit criteria like AIC, BIC, K–S statistic, and log-likelihood. The findings show that fractional calculus offers a potent method for creating new distribution families with increased adaptability and modeling effectiveness. Simulation evidence also confirms superior performance of suggested model across varying scale and shape parameters, thereby justifying its theoretical development and practical adoption. Regression modeling and expansion to multiparameter forms are proposed as future research avenues.