Generalized Chi distribution for non-integer degrees of freedom in modelling biological collectivities

This article aims to expand the practical potential of the Chi and Chi-square distributions by extending their applicability to non-integer degrees of freedom, $k$. The study has two main objectives: first, to explore and leverage the relationship between the Chi, Chi-square, and Gamma distributions to interpret non-integer degrees of freedom; and second, to apply this framework to real survival-time data in the context of biological and health sciences. Three specific cases are analyzed: the germination times of Pinus tropicalis Morelet seeds cultivated in western Cuba; the interval between infection by virulent tuberculous bacilli and death from tuberculosis in guinea pigs; and the period from the onset of symptoms to death due to COVID-19 in patients in Mexico during 2020. A parameter estimation was performed by maximum likelihood (MLE), and the optimal value of $k$ was found to be non-integer in all cases.