Nonlinear weighted least squares fitting Bass diffusion model

The Bass model is one of the most well-known and widely used first-purchase diffusion models in marketing research. The main reason for this is that it finds its origin in a formal theory of product diffusion, and that the model parameters have an easy interpretation in terms of innovation and imitation. Namely, Bass classified adopters (first-time buyers) into two groups: innovators and imitators. Imitators, unlike innovators, are those buyers who are influenced in their adoption by the number of previous buyers. Mathematically, the Bass diffusion model is described by the following differential equation: )] ( )[ ( )] ( [ ) ( t N m t N m q t N m p dt t dN − + − = , , 0 ) 0 ( = N 0 ≥ t , (1) where ) (t N is the cumulative number of adopters of a new product at time t , parameter 0 > m is the total market potential for the new product, and the parameters 0 > p and 0 ≥ q are the coefficients of innovation and imitation, respectively. The adoption rate, dt t dN ) ( , is determined by two additive terms. The first term, )] ( [ t N m p − , represents adoptions due to innovators. The second term, )] ( )[ ( t N m t N m q − , represents adoptions due to imitators. The solution of (1) is given by