A Repeat Purchase Diffusion Model Bayesian Estimation And Control
A Repeat Purchase Diffusion Model Bayesian Estimation And Control
Gary L Lilien
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YA7-^ a ^ 2 a. ) + y. are distributed as bivariate normal with mean (p, ,p«) and covariance matrix 2 -15- The maximum likelihood estimate of N is the mode of the simulated frequency distribution of A/B. Table 1 gives the key pieces of data for two cases of ethical drugs intro- d'jced into two different markets. The data, obtained through a cooperating firm from IMS America, has been disguised by multiplication by an arbitrary constant, to protect company confidentiality. Case 1 is used to valid...ate the model structure and Case 2' to illustrate model use. The parameter esti- mates of the linear model are shown in Table 2, and the distribution of N in Figure 3. Table 3 shows the parameter estimates for the nonlinear model, assuming N = 10, 700, the maximum likelihood estimate, using only the first 12 points for fitting. The function f(t) was modeled as f(t) = 1, t12. This form, consistent with historical decay patterns in the market, works for case 1. Several alternatives were tried (exponential decay, varying times for shift, varying levels for shift) and this one worked adequately both in terms of fit and prediction.
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