Research Article
BibTex RIS Cite

A comparison of methods for estimating parameters of the stochastic Lomax process: through simulation study

Year 2024, Volume: 53 Issue: 2, 495 - 505, 23.04.2024
https://doi.org/10.15672/hujms.1335481

Abstract

In this paper, we study the problem of parameter estimation of the stochastic Lomax diffusion process, this process was introduced in [A. Nafidi, I. Makroz, and R. Gutiérrez Sánchez, A stochastic lomax diffusion process: Statistical inference and application, Mathematics, 2021][14], and the authors suggested the method of simulated annealing to find the maximum likelihood estimators of this process. In this work, we propose alternative methods for finding the maximum likelihood estimators, namely Genetic algorithm and Nelder-Mead, we also investigate the use of Markov Chain Monte Carlo method to determine the model parameters. Finally, an example of application through the simulation of paths for the process is suggested. Then, a comparison is made between the application of three algorithms based on their accuracy and time of execution.

References

  • [1] Y. Aït-Sahalia, Maximum likelihood estimation of discretely sampled diffusions: a closed-form approximation approach, Econometrica 70 (1), 223-262, 2002.
  • [2] F.M. Bandi and P.C. Phillips, Fully nonparametric estimation of scalar diffusion models, Econometrica 71 (1), 241-283, 2003.
  • [3] A. Beskos, O. Papaspiliopoulos, G.O. Roberts and P. Fearnhead, Exact and computationally efficient likelihood-based estimation for discretely observed diffusion processes (with discussion), J R Stat Soc Series B Stat Methodol 68 (3), 33-382, 2006.
  • [4] J. Fan, A selective overview of nonparametric methods in financial econometrics, Stat Sci 20 (4), 317-337, 2005.
  • [5] D. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning, 1989.
  • [6] A. Golightly and D.J. Wilkinson, Bayesian inference for nonlinear multivariate diffusion models observed with error, Comput Stat Data Anal 52 (3), 1674-1693, 2008.
  • [7] R. Gutiérrez, R. Gutiérrez-Sánchez, A. Nafidi and E. Ramos, Three-parameter stochastic lognormal diffusion model: statistical computation and simulating annealingapplication to real case, J. Stat. Comput. Simul. 79 (1), 25-38, 2009.
  • [8] A.S. Hurn and K. Lindsay, Estimating the parameters of stochastic differential equations, Math Comput Simul 48 (4-6), 373-384, 1999.
  • [9] G.J. Jiang and J.L. Knight, A nonparametric approach to the estimation of diffusion processes, with an application to a short-term interest rate model, Econ Theory 13 (5), 615-645, 1997.
  • [10] J.C. Lagarias, J.A. Reeds, M.H. Wright and P.E. Wright, Convergence properties of the neldermead simplex method in low dimensions, SIAM J. Optim. 9 (1), 112-147, 1998.
  • [11] C. Li, Maximum-likelihood estimation for diffusion processes via closed-form density expansions, Ann. Stat. 41 (3), 1350-1380, 2013.
  • [12] A. Nafidi, M. Bahij, B. Achchab and R. Gutiérrez-Sánchez, The stochastic Weibull diffusion process: Computational aspects and simulation, Appl. Math. Comput. 348, 575-587, 2019.
  • [13] A. Nafidi and A. El Azri, A stochastic diffusion process based on the lundqvistkorf growth: Computational aspects and simulation, Math Comput Simul 182, 25-38, 2021.
  • [14] A. Nafidi, I. Makroz and R. Gutiérrez Sánchez, A stochastic lomax diffusion process: Statistical inference and application, Mathematics 9 (1), 100, 2021.
  • [15] M. Plummer, N. Best, K. Cowles and K. Vines,Coda: convergence diagnosis and output analysis for mcmc, R News 6 (1), 711, 2006.
  • [16] M. Plummer, A. Stukalov, M. Denwood and M.M. Plummer, R Package "rjags", Vienna, Austria, 2016.
  • [17] P. Román-Román, D. Romero, M. Rubio and F. Torres-Ruiz, Estimating the parameters of a gompertz-type diffusion process by means of simulated annealing, Appl. Math. Comput. 218 (9), 5121-5131, 2012.
  • [18] P. Román-Román and F. Torres-Ruiz, A stochastic model related to the richards-type growth curve. estimation by means of simulated annealing and variable neighborhood search, Appl. Math. Comput. 266, 579-598, 2015.
  • [19] O. Stramer and M. Bognar, Bayesian inference for irreducible diffusion processes using the pseudo-marginal approach, Bayesian Anal. 6 (2), 231-258, 2011.
  • [20] D. Whitley, A genetic algorithm tutorial, Stat Comput 4 (2), 65-85, 1994.
Year 2024, Volume: 53 Issue: 2, 495 - 505, 23.04.2024
https://doi.org/10.15672/hujms.1335481

Abstract

References

  • [1] Y. Aït-Sahalia, Maximum likelihood estimation of discretely sampled diffusions: a closed-form approximation approach, Econometrica 70 (1), 223-262, 2002.
  • [2] F.M. Bandi and P.C. Phillips, Fully nonparametric estimation of scalar diffusion models, Econometrica 71 (1), 241-283, 2003.
  • [3] A. Beskos, O. Papaspiliopoulos, G.O. Roberts and P. Fearnhead, Exact and computationally efficient likelihood-based estimation for discretely observed diffusion processes (with discussion), J R Stat Soc Series B Stat Methodol 68 (3), 33-382, 2006.
  • [4] J. Fan, A selective overview of nonparametric methods in financial econometrics, Stat Sci 20 (4), 317-337, 2005.
  • [5] D. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning, 1989.
  • [6] A. Golightly and D.J. Wilkinson, Bayesian inference for nonlinear multivariate diffusion models observed with error, Comput Stat Data Anal 52 (3), 1674-1693, 2008.
  • [7] R. Gutiérrez, R. Gutiérrez-Sánchez, A. Nafidi and E. Ramos, Three-parameter stochastic lognormal diffusion model: statistical computation and simulating annealingapplication to real case, J. Stat. Comput. Simul. 79 (1), 25-38, 2009.
  • [8] A.S. Hurn and K. Lindsay, Estimating the parameters of stochastic differential equations, Math Comput Simul 48 (4-6), 373-384, 1999.
  • [9] G.J. Jiang and J.L. Knight, A nonparametric approach to the estimation of diffusion processes, with an application to a short-term interest rate model, Econ Theory 13 (5), 615-645, 1997.
  • [10] J.C. Lagarias, J.A. Reeds, M.H. Wright and P.E. Wright, Convergence properties of the neldermead simplex method in low dimensions, SIAM J. Optim. 9 (1), 112-147, 1998.
  • [11] C. Li, Maximum-likelihood estimation for diffusion processes via closed-form density expansions, Ann. Stat. 41 (3), 1350-1380, 2013.
  • [12] A. Nafidi, M. Bahij, B. Achchab and R. Gutiérrez-Sánchez, The stochastic Weibull diffusion process: Computational aspects and simulation, Appl. Math. Comput. 348, 575-587, 2019.
  • [13] A. Nafidi and A. El Azri, A stochastic diffusion process based on the lundqvistkorf growth: Computational aspects and simulation, Math Comput Simul 182, 25-38, 2021.
  • [14] A. Nafidi, I. Makroz and R. Gutiérrez Sánchez, A stochastic lomax diffusion process: Statistical inference and application, Mathematics 9 (1), 100, 2021.
  • [15] M. Plummer, N. Best, K. Cowles and K. Vines,Coda: convergence diagnosis and output analysis for mcmc, R News 6 (1), 711, 2006.
  • [16] M. Plummer, A. Stukalov, M. Denwood and M.M. Plummer, R Package "rjags", Vienna, Austria, 2016.
  • [17] P. Román-Román, D. Romero, M. Rubio and F. Torres-Ruiz, Estimating the parameters of a gompertz-type diffusion process by means of simulated annealing, Appl. Math. Comput. 218 (9), 5121-5131, 2012.
  • [18] P. Román-Román and F. Torres-Ruiz, A stochastic model related to the richards-type growth curve. estimation by means of simulated annealing and variable neighborhood search, Appl. Math. Comput. 266, 579-598, 2015.
  • [19] O. Stramer and M. Bognar, Bayesian inference for irreducible diffusion processes using the pseudo-marginal approach, Bayesian Anal. 6 (2), 231-258, 2011.
  • [20] D. Whitley, A genetic algorithm tutorial, Stat Comput 4 (2), 65-85, 1994.
There are 20 citations in total.

Details

Primary Language English
Subjects Computational Statistics, Probability Theory, Applied Statistics, Statistics (Other)
Journal Section Statistics
Authors

Ahmed Nafidi 0000-0001-9371-945X

İlyasse Makroz 0000-0002-4216-0657

Early Pub Date March 2, 2024
Publication Date April 23, 2024
Published in Issue Year 2024 Volume: 53 Issue: 2

Cite

APA Nafidi, A., & Makroz, İ. (2024). A comparison of methods for estimating parameters of the stochastic Lomax process: through simulation study. Hacettepe Journal of Mathematics and Statistics, 53(2), 495-505. https://doi.org/10.15672/hujms.1335481
AMA Nafidi A, Makroz İ. A comparison of methods for estimating parameters of the stochastic Lomax process: through simulation study. Hacettepe Journal of Mathematics and Statistics. April 2024;53(2):495-505. doi:10.15672/hujms.1335481
Chicago Nafidi, Ahmed, and İlyasse Makroz. “A Comparison of Methods for Estimating Parameters of the Stochastic Lomax Process: Through Simulation Study”. Hacettepe Journal of Mathematics and Statistics 53, no. 2 (April 2024): 495-505. https://doi.org/10.15672/hujms.1335481.
EndNote Nafidi A, Makroz İ (April 1, 2024) A comparison of methods for estimating parameters of the stochastic Lomax process: through simulation study. Hacettepe Journal of Mathematics and Statistics 53 2 495–505.
IEEE A. Nafidi and İ. Makroz, “A comparison of methods for estimating parameters of the stochastic Lomax process: through simulation study”, Hacettepe Journal of Mathematics and Statistics, vol. 53, no. 2, pp. 495–505, 2024, doi: 10.15672/hujms.1335481.
ISNAD Nafidi, Ahmed - Makroz, İlyasse. “A Comparison of Methods for Estimating Parameters of the Stochastic Lomax Process: Through Simulation Study”. Hacettepe Journal of Mathematics and Statistics 53/2 (April 2024), 495-505. https://doi.org/10.15672/hujms.1335481.
JAMA Nafidi A, Makroz İ. A comparison of methods for estimating parameters of the stochastic Lomax process: through simulation study. Hacettepe Journal of Mathematics and Statistics. 2024;53:495–505.
MLA Nafidi, Ahmed and İlyasse Makroz. “A Comparison of Methods for Estimating Parameters of the Stochastic Lomax Process: Through Simulation Study”. Hacettepe Journal of Mathematics and Statistics, vol. 53, no. 2, 2024, pp. 495-0, doi:10.15672/hujms.1335481.
Vancouver Nafidi A, Makroz İ. A comparison of methods for estimating parameters of the stochastic Lomax process: through simulation study. Hacettepe Journal of Mathematics and Statistics. 2024;53(2):495-50.