An η-Power Stochastic Log-Logistic Diffusion Process: Statistical Computation and Application to Individuals Using the Internet in the United States

A new family of stochastic $\eta$-power log-logistic diffusion processes was introduced and defined based on the classical log-logistic diffusion model. The probabilistic characteristics of the proposed process were derived through an analysis of the associated stochastic differential equation (SDE), including its explicit expressions, the transition probability density function, and the conditional and non-conditional mean functions. The statistical inference of the model was studied, and parameter estimation was performed using the maximum likelihood method based on discrete sampling paths. The proposed probabilistic and statistical framework was applied to data on individuals using the Internet in the United States to assess the practical performance of the model. The empirical results demonstrated that the inclusion of a power in the process improved the goodness of fit compared with the classical formulation, providing better agreement with the observed data. Finally, a small Monte Carlo experiment was performed to examine the robustness of the estimation procedure.