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

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.

[1]  D. Jukic,et al.  On nonlinear weighted total least squares parameter estimation problem for the three-parameter Weibull density , 2010 .

[2]  N. Bailey.  The mathematical theory of epidemics , 1957 .

[3]  Dragan Jukic.  On the existence of the best discrete approximation in lp norm by reciprocals of real polynomials , 2009, J. Approx. Theory.

[4]  Philip E. Gill,et al.  Practical optimization , 1981 .

[5]  Vijay Mahajan,et al.  Chapter 8 New-product diffusion models , 1993, Marketing.

[6]  J. Navarro-Pedreño.  Numerical Methods for Least Squares Problems , 1996 .

[7]  Eugene Demidenko.  On the existence of the least squares estimate in nonlinear growth curve models of exponential type , 1996 .

[8]  Kristian Sabo,et al.  Least‐squares problems for Michaelis–Menten kinetics , 2007 .

[9]  E. Rogers,et al.  Diffusion of innovations , 1964, Encyclopedia of Sport Management.

[10]  Andrej Pázman,et al.  Nonlinear Regression , 2019, Handbook of Regression Analysis With Applications in R.

[11]  Eugene Demidenko,et al.  Criteria for global minimum of sum of squares in nonlinear regression , 2006, Comput. Stat. Data Anal..

[12]  Frank M. Bass,et al.  A New Product Growth for Model Consumer Durables , 2004, Manag. Sci..

[13]  Douglas M. Bates,et al.  Nonlinear Regression Analysis and Its Applications , 1988 .

[14]  Peter Teunissen,et al.  Nonlinear least squares , 1990 .

[15]  Alexander Grey,et al.  The Mathematical Theory of Infectious Diseases and Its Applications , 1977 .

[16]  R. Gunst.  Regression analysis and its application , 1980 .

[17]  R. Kalaba,et al.  Nonlinear Least Squares , 1986 .

[18]  Dragan Jukic,et al.  Total least squares fitting Bass diffusion model , 2011, Math. Comput. Model..

[19]  G. A. Watson,et al.  Use of lp norms in fitting curves and surfaces to data , 2004 .

[20]  N. Ling.  The Mathematical Theory of Infectious Diseases and its applications , 1978 .

[21]  Dragan Jukic,et al.  On nonlinear weighted errors-in-variables parameter estimation problem in the three-parameter Weibull model , 2010, Appl. Math. Comput..

[22]  Dragan Jukić,et al.  Nonlinear weighted least squares estimation of a three-parameter Weibull density with a nonparametric start , 2009 .

[23]  John E. Dennis,et al.  Numerical methods for unconstrained optimization and nonlinear equations , 1983, Prentice Hall series in computational mathematics.

[24]  Vijay Mahajan,et al.  Maximum Likelihood Estimation for an Innovation Diffusion Model of New Product Acceptance , 1982 .

[25]  Charlotte H. Mason,et al.  Technical Note---Nonlinear Least Squares Estimation of New Product Diffusion Models , 1986 .

[26]  R. Scitovski,et al.  Least-squares fitting Gompertz curve , 2004 .

[27]  A. Money,et al.  Nonlinear Lp-Norm Estimation , 2020 .

[28]  Rudolf Scitovski,et al.  Solving parameter estimation problem in new product diffusion models , 2002, Appl. Math. Comput..

[29]  M. Meler,et al.  New product diffusion models , 1995 .

[30]  Vijay Mahajan,et al.  A simple algebraic estimation procedure for innovation diffusion models of new product acceptance , 1986 .

[31]  Rudolf Scitovski,et al.  On the existence of the nonlinear weighted least squares estimate for a three-parameter Weibull distribution , 2008, Comput. Stat. Data Anal..

[32]  V. Mahajan,et al.  Innovation Diffusion Models of New Product Acceptance. , 1987 .

[33]  Dragan Jukic,et al.  On nonlinear weighted least squares estimation of Bass diffusion model , 2013, Appl. Math. Comput..

[34]  E. Demidenko,et al.  Criteria for Unconstrained Global Optimization , 2008 .