In most practical situations, the number of daily responses yi is initially increasing, reaching a peak, and then showing a longer tail dwindling over time, as shown in Figure 1.1(a). However, many researchers have assumed that the daily number of responses yi is a monotonically decreasing function over time, as shown in Figure 1.1(b). They have also considered growth curves that look like a banana-shaped concave function. The growth curves do not fit very well, particularly in postal mail surveys, and Bauer (1991) proposed to arbitrarily exclude the first one or two days (or weeks) to get a better fit. Alternatively, other researchers have assumed that the frequency distribution of yi is symmetrical, as shown in Figure 1.1(c), and have proposed S-shaped logistics or Gompertz curves (Fildes et al. 2008).
Figure 1.1 Frequency distribution of the number of daily responses over time.
Recently, Chun (2012) proposed a geometric response model with two meaningful parameters: (1) an ultimate response rate of the recipients and (2) a delay rate of respondents. His response model with the two parameters has many desirable properties but still has a limitation. The geometric response model is only appropriate for the cases in which the daily number of responses is geometrically decreasing in time, as shown in Figure 1.1(b). In this paper, we extend his model by adding a delivery time to effectively represent the typical S-shaped response pattern in Figure 1.1(a). If the delivery time is negligible, then the response pattern of our model is reduced to the banana-shaped concave function in Figure 1.1(b).
We can imagine many cases in which the processing and delivery time is non-negligible. For example, in postal mail surveys or catalog sales, it takes a longer time to deliver the request to a customer and receive his or her response. In such a case, the delivery time includes the time the postal service takes to deliver a questionnaire (or catalog) to the recipient, the time for a respondent to review and fill out the questionnaire, and the time it takes for responses to get back to the direct marketer.
The response model with a delivery time is called a “heterogeneous starting point” model in Basu, Basu, and Batra (1995), who assume that the delivery time is a uniform (a.k.a., rectangular) distribution. In addition to the uniform distribution, we consider two more probability distributions of delivery time and compare their performances. In the next section, we propose a geometric response model in which the delivery time is expressed in a general form. For a given set of response data, the three parameters in the model can be estimated via the method of maximum likelihood.