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Open AccessArticle

Powers of the Stochastic Gompertz and Lognormal Diffusion Processes, Statistical Inference and Simulation

1
Department of Statistics and Operational Research, Faculty of Science, University of Granada, Avda. Fuentenueva, S/N, 18071 Granada, Spain
2
Department of Mathematics and Informatics, LAMSAD, National School of Applied Sciences Berrechid, University of Hassan 1, Avenue de l’université, BP 280 Berrechid, Morocco
*
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Mathematics 2020, 8(4), 588; https://doi.org/10.3390/math8040588
Received: 5 March 2020 / Revised: 10 April 2020 / Accepted: 13 April 2020 / Published: 15 April 2020
In this paper, we study a new family of Gompertz processes, defined by the power of the homogeneous Gompertz diffusion process, which we term the powers of the stochastic Gompertz diffusion process. First, we show that this homogenous Gompertz diffusion process is stable, by power transformation, and determine the probabilistic characteristics of the process, i.e., its analytic expression, the transition probability density function and the trend functions. We then study the statistical inference in this process. The parameters present in the model are studied by using the maximum likelihood estimation method, based on discrete sampling, thus obtaining the expression of the likelihood estimators and their ergodic properties. We then obtain the power process of the stochastic lognormal diffusion as the limit of the Gompertz process being studied and go on to obtain all the probabilistic characteristics and the statistical inference. Finally, the proposed model is applied to simulated data. View Full-Text
Keywords: powers of stochastic Gompertz diffusion models; powers of stochastic lognormal diffusion models; estimation in diffusion process; stationary distribution and ergodicity; trend function; application to simulated data powers of stochastic Gompertz diffusion models; powers of stochastic lognormal diffusion models; estimation in diffusion process; stationary distribution and ergodicity; trend function; application to simulated data
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Ramos-Ábalos, E.M.; Gutiérrez-Sánchez, R.; Nafidi, A. Powers of the Stochastic Gompertz and Lognormal Diffusion Processes, Statistical Inference and Simulation. Mathematics 2020, 8, 588.

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