• Bass
• bootstrap
• diffusione
• inferenza
• minimi quadrati
• ottimizzazione

The paper presents the main diffusion models and their evolution, focusing on the Bass model, one of the most commonly used first-purchase diffusion models in the marketing research.
After the introduction of the linear and nonlinear least squares and the main unconstrained optimization techniques (such as the univariate search, the direct search methods and the gradient methods), we move forward to describe the confidence intervals using the asymptotic theory, the traditional approach called “the linearization technique” and the most recent modern approach, the bootstrap method.
Furthermore, we focus on problems regarding the best fitting of nonlinear models (as in reality most of the functions are complex and without an analytical solution), its influence on the forecast and we analyze the debate of respectful statisticians who tried to improve or at least to elucidate why the difficulty in searching the global minimum of a regression model and in reducing the uncertainty neighboring the estimators.
As an expansion of all said, an example has been made using the Bass model on the most analyzed datasets in the literature by estimating and exploring the influence of the three parameters (the coefficient of innovation, the coefficient of imitation and the marketing potential) by the number of possessed observations on the prediction of the innovation’s life cycle.
Finally, the paper analyzes the uncertainty neighboring the estimates of the three coefficients by creating the confidence intervals with the traditional approach and the bootstrap method. It can also be visualized the coefficients’ correlation from the point of view of the confidence regions.