A Discrete Bass Model and Its Parameter Estimation

Abstract

Abst,raet A discrete Bass model, which is a discrete analog of the Bass model, is proposed. This discrete Bass rnodel is defined as a difference equation that has an exact solution, The difference equation and the solution respectively t,elld to the differential equation whic:h the Bass model is defined as and the solution when the time interval tends to zero. The discrete Bass model conserves the characterist,ics of the Bass model because the differem, ce equation has am exact solution. Therefore, the discrete Bass model enables us to forecast the innovation diffusion of products and services without a eontinueus-time Bass model. The parameter estimattons of the discrete Bass model are very simple and precise. The difference equation itselfcan be used for the ordinary least squares procedure. Parameter estimation using the ordinary least squares procedure ls equa] Lo t,hat i]sing the nonlinear least squares procedure in the discrete Bass model. 'lihe ordinary least squares procedures iil the discrete Bass model overcome the three shortcomings of the ordinary Ieast squares procedure in the continuous Bass model: t/he time-interval bias, standard error, and multicollinearity.

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“…Consequently, Bass (1969) incorporated the new 'innovator perspective' in the Mansfield model. Bass assumes that diffusion is determined not only by 'imitators' but also by 'innovators' (Satoh, 2001). Thus the Bass specification is also labelled a 'mixed-information-source' model.…”

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“…Generally, according to the model, the innovators and imitators participate in the acceptance of new services, and new users are formed when innovators communicate about their experiences to imitators. The model is mathematically explained by the following formulas [15]: …”

“…or using the analogy with another service for which parameters p and q are known. The latter way, after the introduction of a new service, is to assess p and q parameters by using various mathematical techniques from which the most known are [15]: Ordinary least squares, Maximum likelihood estimation, Non-linear least squares.…”

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“…The exact solution of the above difference equation is given by: (6) Similarly, using the Satoh [15] approach, another discrete equivalent of a continuous model (3) can be obtained. Again, it can be shown that as 0 → δ , m(n) given by (6) and (7) converge to the corresponding continuous SRGM given by (3).…”

“…Two other discrete equivalents of the Kapur and Garg [6] model are also included: these exist in the literature due to Innoue [19] and Satoh [15]. They are subsequently shown to be equivalent to a discrete equivalent of Kapur and Garg [6] that is mathematically and computationally much simpler.…”