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Hyperbolastic Models from a Stochastic Differential Equation Point of View

Author

Listed:
  • Antonio Barrera

    (Departamento de Análisis Matemático, Estadística e Investigación Operativa y Matemática Aplicada, Facultad de Ciencias, Universidad de Málaga, Bulevar Louis Pasteur, 31, 29010 Málaga, Spain
    Instituto de Matemáticas de la Universidad de Granada (IMAG), Calle Ventanilla 11, 18001 Granada, Spain
    These authors contributed equally to this work.)

  • Patricia Román-Román

    (Instituto de Matemáticas de la Universidad de Granada (IMAG), Calle Ventanilla 11, 18001 Granada, Spain
    Departamento de Estadística e Investigación Operativa, Facultad de Ciencias, Universidad de Granada, Avenida Fuente Nueva s/n, 18071 Granada, Spain
    These authors contributed equally to this work.)

  • Francisco Torres-Ruiz

    (Instituto de Matemáticas de la Universidad de Granada (IMAG), Calle Ventanilla 11, 18001 Granada, Spain
    Departamento de Estadística e Investigación Operativa, Facultad de Ciencias, Universidad de Granada, Avenida Fuente Nueva s/n, 18071 Granada, Spain
    These authors contributed equally to this work.)

Abstract

A joint and unified vision of stochastic diffusion models associated with the family of hyperbolastic curves is presented. The motivation behind this approach stems from the fact that all hyperbolastic curves verify a linear differential equation of the Malthusian type. By virtue of this, and by adding a multiplicative noise to said ordinary differential equation, a diffusion process may be associated with each curve whose mean function is said curve. The inference in the resulting processes is presented jointly, as well as the strategies developed to obtain the initial solutions necessary for the numerical resolution of the system of equations resulting from the application of the maximum likelihood method. The common perspective presented is especially useful for the implementation of the necessary procedures for fitting the models to real data. Some examples based on simulated data support the suitability of the development described in the present paper.

Suggested Citation

  • Antonio Barrera & Patricia Román-Román & Francisco Torres-Ruiz, 2021. "Hyperbolastic Models from a Stochastic Differential Equation Point of View," Mathematics, MDPI, vol. 9(16), pages 1-18, August.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:16:p:1835-:d:607965
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    References listed on IDEAS

    as
    1. Antonio Barrera & Patricia Román-Román & Francisco Torres-Ruiz, 2020. "Two Stochastic Differential Equations for Modeling Oscillabolastic-Type Behavior," Mathematics, MDPI, vol. 8(2), pages 1-20, January.
    2. Mohammad A Tabatabai & Jean-Jacques Kengwoung-Keumo & Wayne M Eby & Sejong Bae & Juliette T Guemmegne & Upender Manne & Mona Fouad & Edward E Partridge & Karan P Singh, 2014. "Disparities in Cervical Cancer Mortality Rates as Determined by the Longitudinal Hyperbolastic Mixed-Effects Type II Model," PLOS ONE, Public Library of Science, vol. 9(9), pages 1-18, September.
    3. Kouritzin, Michael A., 2017. "Residual and stratified branching particle filters," Computational Statistics & Data Analysis, Elsevier, vol. 111(C), pages 145-165.
    4. Patricia Román-Román & Juan José Serrano-Pérez & Francisco Torres-Ruiz, 2018. "Some Notes about Inference for the Lognormal Diffusion Process with Exogenous Factors," Mathematics, MDPI, vol. 6(5), pages 1-13, May.
    5. Román-Román, P. & Torres-Ruiz, F., 2015. "A stochastic model related to the Richards-type growth curve. Estimation by means of simulated annealing and variable neighborhood search," Applied Mathematics and Computation, Elsevier, vol. 266(C), pages 579-598.
    6. Rajasekar, S.P. & Pitchaimani, M. & Zhu, Quanxin, 2020. "Progressive dynamics of a stochastic epidemic model with logistic growth and saturated treatment," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 538(C).
    7. Rajasekar, S.P. & Pitchaimani, M., 2020. "Ergodic stationary distribution and extinction of a stochastic SIRS epidemic model with logistic growth and nonlinear incidence," Applied Mathematics and Computation, Elsevier, vol. 377(C).
    8. Luz-Sant’Ana, Istoni & Román-Román, Patricia & Torres-Ruiz, Francisco, 2017. "Modeling oil production and its peak by means of a stochastic diffusion process based on the Hubbert curve," Energy, Elsevier, vol. 133(C), pages 455-470.
    Full references (including those not matched with items on IDEAS)

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