Modelling and forecasting vehicle stocks using the trends of stochastic Gompertz diffusion models: The case of Spain

• Published in: Applied stochastic models in business and industry Vol. 25; no. 3; pp. 385 - 405
• Main Authors: Gutiérrez, R., Gutiérrez-Sánchez, R., Nafidi, A.
• Format: Journal Article
• Language: English
• Published: Chichester, UK John Wiley & Sons, Ltd 01.05.2009
• Subjects: car industry and climate change; stationary distribution; statistical inference with discrete sampling; stochastic homogeneous Gompertz diffusion model; trends analysis; vehicle stocks in Spain
• ISSN: 1524-1904; 1526-4025
• DOI: 10.1002/asmb.754

In the present study, we treat the stochastic homogeneous Gompertz diffusion process (SHGDP) by the approach of the Kolmogorov equation. Firstly, using a transformation in diffusion processes, we show that the probability transition density function of this process has a lognormal time‐dependent distribution, from which the trend and conditional trend functions and the stationary distribution are obtained. Second, the maximum likelihood approach is adapted to the problem of parameters estimation in the drift and the diffusion coefficient using discrete sampling of the process, then the approximated asymptotic confidence intervals of the parameter are obtained. Later, we obtain the corresponding inference of the stochastic homogeneous lognormal diffusion process as limit from the inference of SHGDP when the deceleration factor tends to zero. A statistical methodology, based on the above results, is proposed for trend analysis. Such a methodology is applied to modelling and forecasting vehicle stocks. Finally, an application is given to illustrate the methodology presented using real data, concretely the total vehicle stocks in Spain. Copyright © 2008 John Wiley & Sons, Ltd.