The Bass model is one of the most well-known and widely used first-purchase diffusion models in marketing research. Estimation of its parameters has been approached in the literature by various techniques. In this paper, we consider the parameter estimation approach for the Bass model based on nonlinear weighted least squares fitting of its derivative known as the adoption curve. We show that it is possible that the least squares estimate does not exist. As a main result, two theorems on the existence of the least squares estimate are obtained, as well as their generalization in the l_{s} norm (1 ≤ s < ∞). One of them gives necessary and sufficient conditions which guarantee the existence of the least squares estimate. Several illustrative numerical examples are given to support the theoretical work.

# International Journal of Applied Mathematics and Computer Science

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# On parameter estimation in the bass model by nonlinear least squares fitting the adoption curve

#### Open Access

Keywords : Bass model; least squares estimate; existence problem; data fitting

Atieg, A. and Watson, G.A. (2004). Use of

*l*norms in fitting curves and surfaces to data,_{p}*The ANZIAM Journal***45**(E): C187-C200.Google ScholarBailey, N.T.J. (1975).

*The Mathematical Theory of Infectious**Diseases and Its Applications*, Griffin, London.Google ScholarBailey, N.T.J. (1957).

*The Mathematical Theory of Epidemics*, Griffin, London.Google ScholarBass, F.M. (1969). A new product growth model for consumer durables,

*Management Science***15**(5): 215-227.CrossrefGoogle ScholarBates, D.M. andWatts, D.G. (1988).

*Nonlinear Regression Analysis**and Its Applications*, Wiley, New York, NY.Google ScholarBjörck, Å. (1996).

*Numerical Methods for Least Squares Problems*, SIAM, Philadelphia, PA.Google ScholarDemidenko, E.Z. (2008). Criteria for unconstrained global optimization,

*Journal of Optimization Theory and Applications***136**(3): 375-395.Google ScholarDemidenko, E.Z. (2006). Criteria for global minimum of sum of squares in nonlinear regression,

*Computational Statistics**& Data Analysis***51**(3): 1739-1753.CrossrefGoogle ScholarDemidenko, E.Z. (1996). On the existence of the least squares estimate in nonlinear growth curve models of exponential type,

*Communications in Statistics-Theory and Methods***25**(1): 159-182.CrossrefGoogle ScholarDennis, J.E. and Schnabel, R.B. (1996).

*Numerical Methods**for Unconstrained Optimization and Nonlinear Equations*, SIAM, Philadelphia, PA.Google ScholarGill, P.E., Murray, W. and Wright, M.H. (1981).

*Practical Optimization*, Academic Press, London.Google ScholarGonin, R. and Money, A.H. (1989).

*Nonlinear L*, Marcel Dekker, New York, NY.Google Scholar_{p}-Norm EstimationHadeler, K.P., Jukić, D. and Sabo, K. (2007). Least squares problems for Michaelis Menten kinetics,

*Mathematical**Methods in the Applied Sciences***30**(11): 1231-1241.Web of ScienceGoogle ScholarJukić, D. (2011). Total least squares fitting Bass diffusion model,

*Mathematical and Computer Modelling***53**(9-10): 1756-1770.Web of ScienceGoogle ScholarJukić, D. (2013) On nonlinear weighted least squares estimation of Bass diffusion model,

*Applied Mathematics and Computation*, (accepted).Web of ScienceGoogle ScholarJukić, D. and Marković, D. (2010). On nonlinear weighted errors-in-variables parameter estimation problem in the three-parameter Weibull model,

*Applied Mathematics and**Computation***215**(10): 3599-3609.Google ScholarJukić, D. (2009). On the existence of the best discrete approximation in

*l*norm by reciprocals of real polynomials,_{p}*Journal of Approximation Theory***156**(2): 212-222.Web of ScienceGoogle ScholarJukić, D., Benšić, M. and Scitovski, R. (2008). On the existence of the nonlinear weighted least squares estimate for a three-parameter Weibull distribution,

*Computational**Statistics & Data Analysis***52**(9): 4502-4511.Google ScholarJukić, D., Kralik, G. and Scitovski, R. (2004). Least squares fitting Gompertz curve,

*Journal of Computational and Applied**Mathematics***169**(2): 359-375.Google ScholarMahajan, V. Muller, E. and Wind, Y. (Eds.). (2000).

*New-**Product Diffusion Models*, Kluwer Academic Publishers, London.Google ScholarMahajan, V., Mason, C.H. and Srinivasan, V. (1986). An evaluation of estimation procedures for new product diffusion models,

*in*V. Mahajan and Y. Wind (Eds.),*Innovation**Diffusion Models of New Product Acceptance*, Ballinger Publishing Company, Cambridge, pp. 203-232.Google ScholarMahajan, V. and Sharma, S. (1986). A simple algebraic estimation procedure for innovation diffusion models of new product acceptance,

*Technological Forecasting and**Social Change***30**(4): 331-346.Google ScholarMarković, D. and Jukić, D. (2010). On nonlinear weighted total least squares parameter estimation problem for the three-parameter Weibull density,

*Applied Mathematical**Modelling***34**(7): 1839-1848.Google ScholarMarković, D., Jukić, D. and Benšić, M. (2009). Nonlinear weighted least squares estimation of a three-parameter Weibull density with a nonparametric start,

*Journal**of Computational and Applied Mathematics***228**(1): 304-312.CrossrefGoogle ScholarRogers, E.M. (1962).

*Diffusion of Innovations*, The Free Press, New York, NY.Google ScholarRoss, G.J.S. (1990).

*Nonlinear Estimation*, Springer, New York, NY.Google ScholarSchmittlein, D. and Mahajan, V. (1982). Maximum likelihood estimation for an innovation diffusion model of new product acceptance,

*Marketing Science***1**(1): 57-78.CrossrefGoogle ScholarScitovski, R. and Meler, M. (2002). Solving parameter estimation problem in new product diffusion models,

*Applied**Mathematics and Computation***127**(1): 45-63.Google ScholarSeber, G.A.F. and Wild, C.J. (1989).

*Nonlinear Regression*, Wiley, New York, NY.Google ScholarSrinivasan, V. and Mason, C.H. (1986). Nonlinear least squares estimation of new product diffusion models,

*Marketing**Science***5**(2): 169-178.Google Scholar

## About the article

**Published Online**: 2013-03-26

**Published in Print**: 2013-03-01

**Citation Information: **International Journal of Applied Mathematics and Computer Science, ISSN (Online) 2083-8492, ISSN (Print) 1641-876X, DOI: https://doi.org/10.2478/amcs-2013-0012.

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