A Stochastic Diffusion Process Based on the Two-Parameters Weibull Density Function

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.

Authors:

• Meriem Bahij
• Ahmed Nafidi
• Boujemâa Achchab
• Sílvio M. A. Gama
• José A. O. Matos

Keywords:

• Diffusion process
• discrete sampling
• likelihood estimation method
• simulation
• stochastic diffusion equation
• trends functions
• bi-parameters Weibull density function.

References:

[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.