This paper introduces a novel non-homogeneous stochastic diffusion process, useful for modeling both decreasing and increasing trend data. The model is based on a generalized Gamma-like curve. We derive the probabilistic characteristics of the proposed process, including a closed-form unique solution to the stochastic differential equation, the transition probability density function, and both conditional and unconditional trend functions. The process parameters are estimated using the maximum likelihood (ML) method with discrete sampling paths. A small Monte Carlo experiment is conducted to evaluate the finite sample behavior of the trend function. The practical utility of the proposed process is demonstrated by fitting it to two real-world data sets, one exhibiting a decreasing trend and the other an increasing trend.
Citation: Safa' Alsheyab, Mohammed K. Shakhatreh. A new stochastic diffusion process based on generalized Gamma-like curve: inference, computation, with applications[J]. AIMS Mathematics, 2024, 9(10): 27687-27703. doi: 10.3934/math.20241344
This paper introduces a novel non-homogeneous stochastic diffusion process, useful for modeling both decreasing and increasing trend data. The model is based on a generalized Gamma-like curve. We derive the probabilistic characteristics of the proposed process, including a closed-form unique solution to the stochastic differential equation, the transition probability density function, and both conditional and unconditional trend functions. The process parameters are estimated using the maximum likelihood (ML) method with discrete sampling paths. A small Monte Carlo experiment is conducted to evaluate the finite sample behavior of the trend function. The practical utility of the proposed process is demonstrated by fitting it to two real-world data sets, one exhibiting a decreasing trend and the other an increasing trend.
| [1] |
A. El Azri, N. Ahmed, A stochastic log-logistic diffusion process: statistical computational aspects and application to real data, Stoch. Models, 40 (2024), 261–277. https://doi.org/10.1080/15326349.2023.2241070 doi: 10.1080/15326349.2023.2241070
|
| [2] |
A. Nafidi, A. El Azri, R. Gutiérrez-Sánchez, A stochastic Schumacher diffusion process: probability characteristics computation and statistical analysis, Methodol. Comput. Appl. Probab., 25 (2023), 66. https://doi.org/10.1007/s11009-023-10031-4 doi: 10.1007/s11009-023-10031-4
|
| [3] |
A. Nafidi, I. Makroz, R. Gutiérrez-Sánchez, A stochastic Lomax diffusion process: statistical inference and application, Mathematics, 9 (2021), 100. https://doi.org/10.3390/math9010100 doi: 10.3390/math9010100
|
| [4] |
A. Nafidi, M. Bahij, R. Gutiérrez-Sánchez, B. Achchab, Two-parameter stochastic Weibull diffusion model: statistical inference and application to real modeling example, Mathematics, 8 (2020), 160. https://doi.org/10.3390/math8020160 doi: 10.3390/math8020160
|
| [5] |
A. Nafidi, G. Moutabir, R. Gutiérrez-Sánchez, E. Ramos-Ábalos, Stochastic square of the Brennan-Schwartz diffusion process: statistical computation and application, Methodol. Comput. Appl. Probab., 7 (2020), 455–476. https://doi.org/10.1007/s11009-019-09714-8 doi: 10.1007/s11009-019-09714-8
|
| [6] |
A. Nafidi, G. Moutabir, R. Gutiérrez-Sánchez, Stochastic Brennan–Schwartz diffusion process: statistical computation and application, Mathematics, 7 (2019), 1062. https://doi.org/10.3390/math7111062 doi: 10.3390/math7111062
|
| [7] |
R. Gutiérrez, R. Gutiérrez-Sánchez, A. Nafidi, The trend of the total stock of the private car-petrol in Spain: stochastic modelling using a new gamma diffusion process, Appl. Energy, 86 (2009), 18–24. https://doi.org/10.1016/j.apenergy.2008.03.016 doi: 10.1016/j.apenergy.2008.03.016
|
| [8] |
R. Gutiérrez, R. Gutiérrez-Sánchez, A. Nafidi, Modelling and forecasting vehicle stocks using the trends of stochastic Gompertz diffusion models: the case of Spain, Appl. Stoch. Model. Bus. Ind., 25 (2009), 385–405. https://doi.org/10.1002/asmb.754 doi: 10.1002/asmb.754
|
| [9] |
R. Gutiérrez, R. Gutiérrez-Sánchez, A. Nafidi, The stochastic Rayleigh diffusion model: statistical inference and computational aspects. Applications to modelling of real cases, Appl. Math. Comput., 175 (2006), 628–644. https://doi.org/10.1016/j.amc.2005.07.047 doi: 10.1016/j.amc.2005.07.047
|
| [10] |
B. M. Bibby, M. Sørensen, Martingale estimation functions for discretely observed diffusion processes, Bernoulli, 1 (1995), 17–39. https://doi.org/10.2307/3318679 doi: 10.2307/3318679
|
| [11] | P. E. Kloeden, E. Platen, Numerical solution of stochastic differential equations, Springer Berlin, Heidelberg, 1992. https://doi.org/10.1007/978-3-662-12616-5 |
| [12] | B. L. S. Prakasa Rao, Statistical inference for diffusion type processes, Arnold, London, UK, 1999. |
| [13] |
E. W. Stacy, A generalization of the Gamma distribution, Ann. Math. Statist., 33 (1962), 1187–1192. https://doi.org/10.1214/aoms/1177704481 doi: 10.1214/aoms/1177704481
|
| [14] | M. J. Schervish, Theory of statistics, Springer-Verlag, New York, USA, 1995. https://doi.org/10.1007/978-1-4612-4250-5 |
| [15] | The R Core Team, R: A language and environment for statistical computing, R Foundation for Statistical Computing, Vienna, Austria, 2016. Available from: https://web.mit.edu/r_v3.3.1/fullrefman.pdf. |
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Safa' Alsheyab, Mohammed K. Shakhatreh. A new stochastic diffusion process based on generalized Gamma-like curve: inference, computation, with applications[J]. AIMS Mathematics, 2024, 9(10): 27687-27703. doi: 10.3934/math.20241344
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