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.
Citation: Juan Carlos Castro-Palacio, Pedro Fernández de Córdoba, Álvaro González-Cortés, J. M. Isidro, A. Noverques, Marcos Orellana-Panchame, Sarira Sahu, Enrique A. Sánchez-Pérez. Generalized Chi distribution for non-integer degrees of freedom in modelling biological collectivities[J]. AIMS Mathematics, 2025, 10(10): 25033-25048. doi: 10.3934/math.20251109
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.
| [1] |
D. W. K. Andrews, Chi-square diagnostic tests for econometric models, J. Econometrics, 37 (1988), 135–156. https://doi.org/10.1016/0304-4076(88)90079-6 doi: 10.1016/0304-4076(88)90079-6
|
| [2] | K. Pearson, On the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling, London, Edinburgh, Dublin Philos. Mag. J. Sci., 50 (1900), 157–175. |
| [3] | E. J. Gumbel, On the reliability of the classical chi-square test, Ann. Math. Stat., 14 (1943), 253–263. |
| [4] |
R. A. Fisher, The conditions under which chi-squared measures the discrepancey between observation and hypothesis, J. R. Stat. Soc., 87 (1924), 442–450. https://doi.org/10.1093/bjps/34.2.105 doi: 10.1093/bjps/34.2.105
|
| [5] | B. McWilliams, M. M. Newmann, D. Sprevak, The probability distribution of wind velocity and direction, Wind Eng., 3 (1979), 269–273. |
| [6] | R. C. Tolman, The principles of statistical mechanics, New York: Dover Publications, 1980. |
| [7] | C. W. Zhang, M. Xie, J. Y. Liu, T. N. Goh, A control chart for the gamma distribution as a model of time between events, Int. J. Prod. Res., 45 (2007), 5649–5666. |
| [8] | D. W. Coit, T. Jin, Gamma distribution parameter estimation for field reliability data with missing failure times, IIE Trans., 32 (2000), 1161–1166. |
| [9] | R. Winkelmann, A count data model for gamma waiting times, Stat. Pap., 37 (1996), 177–187. |
| [10] | J. C. Castro-Palacio, J. M. Isidro, E. Navarro-Pardo, L. Velázquez-Abad, P. Fernández-de Córdoba, Monte carlo simulation of a modified chi distribution with unequal variances in the generating gaussians. A discrete methodology to study collective response times, Mathematics, 9 (2020), 77. |
| [11] | N. Ortigosa, M. Orellana-Panchame, J. C. Castro-Palacio, P. Córdoba, J. M. Isidro, Monte carlo simulation of a modified chi distribution considering asymmetry in the generating functions: Application to the study of health-related variables, Symmetry, 13 (2021), 924. |
| [12] | M. Nakagami, The m-distribution—a general formula of intensity distribution of rapid fading, In: Statistical methods in radio wave propagation, Belfast: The Universities Press, 1960, 3–36. https://doi.org/10.1016/B978-0-08-009306-2.50005-4 |
| [13] | D. D. Wackerly, W. Mendenhall, R. L. Scheaffer, Mathematical statistics with applications, 7 Eds., Belmont: Thomson Higher Education, 2008. |
Figures(5) / Tables(6)
Juan Carlos Castro-Palacio, Pedro Fernández de Córdoba, Álvaro González-Cortés, J. M. Isidro, A. Noverques, Marcos Orellana-Panchame, Sarira Sahu, Enrique A. Sánchez-Pérez. Generalized Chi distribution for non-integer degrees of freedom in modelling biological collectivities[J]. AIMS Mathematics, 2025, 10(10): 25033-25048. doi: 10.3934/math.20251109
DownLoad: