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 ls 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.
If the inline PDF is not rendering correctly, you can download the PDF file here.
Atieg A. and Watson G.A. (2004). Use of lpnorms in fitting curves and surfaces to data The ANZIAM Journal 45(E): C187-C200.
Bailey N.T.J. (1975). The Mathematical Theory of InfectiousDiseases and Its Applications Griffin London.
Bailey N.T.J. (1957). The Mathematical Theory of Epidemics Griffin London.
Bass F.M. (1969). A new product growth model for consumer durables Management Science 15(5): 215-227.
Bates D.M. andWatts D.G. (1988). Nonlinear Regression Analysisand Its Applications Wiley New York NY.
Björck Å. (1996). Numerical Methods for Least Squares Problems SIAM Philadelphia PA.
Demidenko E.Z. (2008). Criteria for unconstrained global optimization Journal of Optimization Theory and Applications136(3): 375-395.
Demidenko E.Z. (2006). Criteria for global minimum of sum of squares in nonlinear regression Computational Statistics& Data Analysis 51(3): 1739-1753.
Demidenko E.Z. (1996). On the existence of the least squares estimate in nonlinear growth curve models of exponential type Communications in Statistics-Theory and Methods25(1): 159-182.
Dennis J.E. and Schnabel R.B. (1996). Numerical Methodsfor Unconstrained Optimization and Nonlinear Equations SIAM Philadelphia PA.
Gill P.E. Murray W. and Wright M.H. (1981). Practical Optimization Academic Press London.
Gonin R. and Money A.H. (1989). Nonlinear Lp-Norm Estimation Marcel Dekker New York NY.
Hadeler K.P. Jukić D. and Sabo K. (2007). Least squares problems for Michaelis Menten kinetics MathematicalMethods in the Applied Sciences 30(11): 1231-1241.
Jukić D. (2011). Total least squares fitting Bass diffusion model Mathematical and Computer Modelling 53(9-10): 1756-1770.
Jukić D. (2013) On nonlinear weighted least squares estimation of Bass diffusion model Applied Mathematics and Computation (accepted).
Jukić D. and Marković D. (2010). On nonlinear weighted errors-in-variables parameter estimation problem in the three-parameter Weibull model Applied Mathematics andComputation 215(10): 3599-3609.
Jukić D. (2009). On the existence of the best discrete approximation in lpnorm by reciprocals of real polynomials Journal of Approximation Theory 156(2): 212-222.
Jukić D. Benšić M. and Scitovski R. (2008). On the existence of the nonlinear weighted least squares estimate for a three-parameter Weibull distribution ComputationalStatistics & Data Analysis 52(9): 4502-4511.
Jukić D. Kralik G. and Scitovski R. (2004). Least squares fitting Gompertz curve Journal of Computational and AppliedMathematics 169(2): 359-375.
Mahajan V. Muller E. and Wind Y. (Eds.). (2000). New-Product Diffusion Models Kluwer Academic Publishers London.
Mahajan 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.) InnovationDiffusion Models of New Product Acceptance Ballinger Publishing Company Cambridge pp. 203-232.
Mahajan V. and Sharma S. (1986). A simple algebraic estimation procedure for innovation diffusion models of new product acceptance Technological Forecasting andSocial Change 30(4): 331-346.
Marković D. and Jukić D. (2010). On nonlinear weighted total least squares parameter estimation problem for the three-parameter Weibull density Applied MathematicalModelling 34(7): 1839-1848.
Marković D. Jukić D. and Benšić M. (2009). Nonlinear weighted least squares estimation of a three-parameter Weibull density with a nonparametric start Journalof Computational and Applied Mathematics 228(1): 304-312.
Rogers E.M. (1962). Diffusion of Innovations The Free Press New York NY.
Ross G.J.S. (1990). Nonlinear Estimation Springer New York NY.
Schmittlein D. and Mahajan V. (1982). Maximum likelihood estimation for an innovation diffusion model of new product acceptance Marketing Science 1(1): 57-78.
Scitovski R. and Meler M. (2002). Solving parameter estimation problem in new product diffusion models AppliedMathematics and Computation 127(1): 45-63.
Seber G.A.F. and Wild C.J. (1989). Nonlinear Regression Wiley New York NY.
Srinivasan V. and Mason C.H. (1986). Nonlinear least squares estimation of new product diffusion models MarketingScience 5(2): 169-178.