Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 30855
A Stochastic Diffusion Process Based on the Two-Parameters Weibull Density Function

Authors: Meriem Bahij, Ahmed Nafidi, Boujemâa Achchab, Sílvio M. A. Gama, José A. O. Matos


Stochastic modeling concerns the use of probability to model real-world situations in which uncertainty is present. Therefore, the purpose of stochastic modeling is to estimate the probability of outcomes within a forecast, i.e. to be able to predict what conditions or decisions might happen under different situations. In the present study, we present a model of a stochastic diffusion process based on the bi-Weibull distribution function (its trend is proportional to the bi-Weibull probability density function). In general, the Weibull distribution has the ability to assume the characteristics of many different types of distributions. This has made it very popular among engineers and quality practitioners, who have considered it the most commonly used distribution for studying problems such as modeling reliability data, accelerated life testing, and maintainability modeling and analysis. In this work, we start by obtaining the probabilistic characteristics of this model, as the explicit expression of the process, its trends, and its distribution by transforming the diffusion process in a Wiener process as shown in the Ricciaardi theorem. Then, we develop the statistical inference of this model using the maximum likelihood methodology. Finally, we analyse with simulated data the computational problems associated with the parameters, an issue of great importance in its application to real data with the use of the convergence analysis methods. Overall, the use of a stochastic model reflects only a pragmatic decision on the part of the modeler. According to the data that is available and the universe of models known to the modeler, this model represents the best currently available description of the phenomenon under consideration.

Keywords: Simulation, diffusion process, discrete sampling, bi-parameters weibull density function, likelihood estimation method, stochastic diffusion equation, trends functions

Digital Object Identifier (DOI):

Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1198


[1] R. F. Woolson and W. R. Clarke, Statistical Methods for the Analysis of Biomedical Data, 2nd ed. John Wiley & Sons, Vol.371, New York, United States, 2000.
[2] R. L. Mason, R. F. Gunst, and J. L. Hess Statistical Design and Analysis of Experiments: with Applications to Engineering and Science,Wiley, New York, United States, 1989.
[3] W. R. Blischke and D. N. P. Murthy, Probability distributions for modeling time to failure, in Reliability: Modeling, Prediction, and Optimization, John Wiley & Sons, Inc.,Hoboken, NJ, USA, 2000.
[4] S. A. Klugman, and R. Parsa, Fitting bivariate loss distributions with copulas, Insurance: Mathematics and Economics, Elsevier, Vol. 24, no.1, 1999, pp. 139–148.
[5] D. J. Davis, An Analysis of some Failure Data, Journal of the American Statistical Association, Taylor & Francis Group, Vol. 47, no.250, 1952, pp. 113–150.
[6] P. Feigl and M. Zelen, Estimation of exponential survival probabilities with concomitant information, Biometrics, JSTOR, 1965, pp. 826–838.
[7] D. R Cox, Renewal Theory Methuen, CoxRenewal Theory1962, London, 1962.
[8] E. J. Gumbel, Statistics of extremes. 1958, Columbia Univ. press, New York, 1958.
[9] J. Lieblein and M. Zelen, Statistical investigation of the fatigue life of deep-groove ball bearings, Journal of Research of the National Bureau of Standards, Citeseer, Vol. 57, no.5, 1956, pp. 273–316.
[10] M. C. Pike, A method of analysis of a certain class of experiments in carcinogenesis, Biometrics, JSTOR, Vol. 22, no.1, 1966, pp. 142–161.
[11] J. W. Boag, Maximum Likelihood Estimates of the Proportion of Patients Cured by Cancer Therapy, Journal of the Royal Statistical Society. Series B (Methodological), Royal Statistical Society, Wiley, Vol. 11, no.1, 1949, pp. 15–53.
[12] A. N. Giovanis and C. H. Skiadas, A Stochastic Logistic Innovation Diffusion Model Studying the Electricity Consumption in Greece and the United States, Technological Forecasting and Social Change, Vol. 61, 1999, pp. 235–246.
[13] A. Katsamaki and C. H. Skiadas, Analytic solution and estimation of parameters on stochastics exponential model for a technological diffusion process, Applied Stochastics Model and Data Analysis, Vol. 11, 1995, pp. 59–75.
[14] C. Skiadas and A. Giovani, A stochastic bass innovation diffusion model for studying the growth of electricity consumption in Greece, Applied Stochastic Models and Data Analysis, Vol. 13, 1997, pp. 85–101.
[15] R. Gutie´rrez-Sa´nchez, A. Nafidi, A. Pascual, E. R. A´ balos, Three parameter gamma-type growth curve, using a stochastic gamma diffusion model: Computational statistical aspects and simulation, Mathematics and Computers in Simulation, Vol. 82, 2011, pp. 234–243.
[16] R. Guti´errez, R. Guti´errez-S´anchez, A. Nafidi and E. Ramos, A diffusion model with cubic drift: statistical and computational aspects and application to modeling of the global CO2 emission in Spain, Environmetrics, Vol. 18, 2007, pp. 55–69.
[17] R. Guti´errez, R. Guti´errez-S´anchez, A. Nafidi and E. Ramos, Studying the vehicule park in Spain using the lognormal and Gompertz diffusion processes, Proceedings od SEIO’04, Vol. 18, 2004, pp. 171–172.
[18] A. V. Egorov, H. Li, and Y. Xu, Maximum likelihood estimation of time-inhomogeneous diffusions, Journal of Econometric, Vol. 114, 2003, pp. 107–139.
[19] Y. Ait-Sahalia, R. Kimmel, Maximum likelihood estimation of stochastic volatility models, Journal of Financial Economics, Vol. 83, 2007, pp. 413–452.
[20] F. Casas, Solution of linear partial differential equations by Lie algebraic methods, Journal of Computational and Applied Mathematics, Vol. 76, 1996, pp. 159–170.
[21] P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer-Verlag, Applications of Mathematics Series, no.23, 1991.
[22] LM. Ricciardi, Diffusion processes and related topics in biology. Lecture notes in biomathematics, Springer-Verlag, Berlin, 1977.
[23] P. W. Zenha, Invariance of Maximum Likelihood Estimators, The Annals of Mathematical Statistics, Ann. Math. Statist., Vol. 37, no.3, 1966, pp. 744.