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2013 | 23 | 1 | 145-155
Tytuł artykułu

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

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Rocznik
Tom
23
Numer
1
Strony
145-155
Opis fizyczny
Daty
wydano
2013
otrzymano
2012-02-07
poprawiono
2012-08-06
Twórcy
  • Department of Mathematics, University of Osijek, Trg Ljudevita Gaja 6, HR-31 000 Osijek, Croatia,
  • Department of Mathematics, University of Osijek, Trg Ljudevita Gaja 6, HR-31 000 Osijek, Croatia,
Bibliografia
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  • Dennis, J.E. and Schnabel, R.B. (1996). Numerical Methods for Unconstrained Optimization and Nonlinear Equations, SIAM, Philadelphia, PA.
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  • 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 and Computation 215(10): 3599-3609.
  • Jukić, D. (2009). On the existence of the best discrete approximation in lp norm 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, Computational Statistics & Data Analysis 52(9): 4502-4511.
  • Jukić, D., Kralik, G. and Scitovski, R. (2004). Least squares fitting Gompertz curve, Journal of Computational and Applied Mathematics 169(2): 359-375.
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  • Mahajan, 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.
  • Marković, 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.
  • Marković, 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.
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Typ dokumentu
Bibliografia
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Identyfikator YADDA
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