Journal
APPLIED STOCHASTIC MODELS IN BUSINESS AND INDUSTRY
Volume 25, Issue 3, Pages 385-405Publisher
WILEY
DOI: 10.1002/asmb.754
Keywords
stochastic homogeneous Gompertz diffusion model; statistical inference with discrete sampling; stationary distribution; trends analysis; vehicle stocks in Spain; car industry and climate change
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Funding
- Ministerio de Educacion y Ciencia [MTM 2005-09209, MTM-2008-05785]
- Junta de Andalucia [P06-FQM-02271]
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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 (C) 2008 John Wiley & Sons, Ltd.
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