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Suppose that a survey form, catalog, or solicitation letter is sent to N customers in the selected group, and their responses are recorded over time. Let y={y1, y2, ..., yk} denote the number of responses received during each of the past k days (or weeks) after the launch of the direct marketing campaign. For notational convenience, let si = y1 + y2 +...+ yi be the total number of responses accumulated by the end of the i th day. The cumulative number of responses si is usually a monotonically increasing function of time i.

Many researchers have proposed various types of growth curves and considered different methods of estimating the model parameters. For example, Huxley (1980) made the first formal attempt to model the response pattern of a mail survey by using the following equation:

Equation (1-1)


where α (>0) and β (<1) are unknown parameters to be estimated empirically and N is the number of questionnaires mailed initially. The growth curve of the response rate is similar to the cumulative distribution of an exponential probability distribution:

Equation (1-2)


where α=α/N and β=– ln (β). After a log-transformation, the growth curve in (1-1) can be written as a simple linear regression model,

Equation (1-3)


from which he found the least square estimators of α and β for given data.

Huxley (1980) mailed out N =4,314 questionnaires initially and recorded the cumulative number of questionnaires {s1, s2, ..., s17} received during the 17-week period. Note that, in his response model in (1), he implicitly assumed that si approaches N as i increases to infinity, which implies that all questionnaires will be returned eventually. When i =0, on the other hand, the cumulative number of responses s0 has a nonzero value.

Since Huxley’s pioneering work, numerous researchers have modified his original model or proposed alternative ones (e.g., Hill 1981; McGowan 1986; Bauer 1987, 1991; Wilson and Singer 1991; Basu, Basu, and Batra 1995; Pan 2010; Chun 2012). Most response models have two or three parameters, whereas McGowan (1986) proposed a logistics curve that has five unknown parameters that have no meaningful interpretations.

In general, the customer response models are classified into (1) the growth curve model and (2) the probabilistic response model. First, most of the earlier research focused on how to find the best growth curve that fits a given response data (Huxley 1980; Hill 1981; McGowan 1986; Bauer 1987, 1991). The method of least squares is usually used to estimate the parameter values. Second, in the probabilistic response model, the daily response of each respondent is modeled as a Bernoulli process so that the total responses in each day can be a random variable from a geometric distribution. In such a case, the model parameters are estimated by the maximum likelihood method (Wilson and Singer 1991; Chun 2012).

The need for an accurate response model is significant in direct marketing. Based on the customer response rate and speed, a direct marketer can adjust the marketing campaign, the message, or the target population to identify the most likely responders and improve the return on investment. Finn (1983) concluded that “more research into the nature of response functions in mail surveys is needed. If a consistently accurate predictive technique can be found, it will be invaluable to users of mail surveys.”

In the following sections, we propose a new probabilistic response model that has many desirable properties. First, the cumulative number of responses is si=0 when i=0, and has an asymptote si < N when i=∞. Second, the response model is flexible enough to represent various types of response patterns with different shapes and locations. Third, the response model is parsimonious, with a smaller number of parameters. Fourth, each of the model’s parameters has a meaningful interpretation. Few researchers have proposed response models that have all four of these desirable properties.

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