Statistical inference in one-dimensional stochastic ...

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Statistical inference in one-dimensional stochastic ...

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Statistical inference in one-dimensional stochastic. Gamma diffusion process with external information. R. GutiÃ©rrez, A. Nafidi1, R. GutiÃ©rrez-SÃ¡nchez and S.

Tenth International Conference Zaragoza-Pau on Applied Mathematics and Statistics Jaca, September 15–17th 2008

Statistical inference in one-dimensional stochastic Gamma diffusion process with external information R. Guti´ errez, A. Nafidi1 , R. Guti´ errez-S´ anchez and S. Melliani2 SUMMARY In this work, we consider a new extension of the one -dimensional stochastic gamma diffusion process (cf. [2]) by introducing external time functions as exogenous factors, in the same way the exogenous factors have been introduced in the lognormal and Gompertz processes (cf. [3]), Gompertz process (cf. [1] ) and Vasicek process (cf. [4]). Firstly, we determine the probabilistic characteristics of the studied process as the analytical expression of the process, its transition probability density function and their trend functions. Secondly, we study the statistical inference in this process: we estimate the parameters present in the model by using the maximum likelihood estimation method in basis of the discreet sampling of the process, thus obtaining the expression of the likelihood estimators and their properties (statistical distributions, sufficiency and completeness), then, the confidence intervals of the parameters. Keywords: Stochastic Gamma diffusion process, Exogenous factors, Statistical inference using discreet sampling. AMS Classification:

60J60, 62M05

References [1] G. Albano and V. Giorno. A stochastic model in tumor growth. Journal of Theoretical Biology 242(2), 329-336, 2006. ´rrez, R. Gutie ´rrez-Sa ńchez and A. Nafidi. The trend of the total stock [2] R. Gutie of the private car-petrol in Spain: Stochastic modelling using a new gamma diffusion process. Applied Energy doi:10.1016/j.apenergy.2008.03.016, 2008. ´rrez, R. Gutie ´rrez-Sa ńchez and A. Nafidi. Electricity consumption [3] R. Gutie in Morocco: Stochastic Gompertz exogenous factors diffusion analysis. Applied Energy 83,1139–1151, 2006. ´rrez, A. Nafidi, R. Gutie ´rrez-Sa ńchez R and S. Melliani. A non[4] R. Gutie homogenous stochastic Vasicek diffusion model: Application to global CO2 emission in Marocco. In MAMERN’07, B. Amaziane, D. Barrera,. . . (eds.), pp. 323–329, Universidad de Granada, Granada, 2007. 1

Department of Statistics and Operational Research University of Granada Facultad de Ciencias, Campus de Fuentenueva 18071 Granada, Spain [email protected]

2

Department of Applied Mathematics and Informatics University of Sultan Mouly Slimane Faculté des Sciences et Techniques B.P: 523, Béni Mellal, Morocco [email protected]