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Customer Response Model

Suppose that we send out a request to N individuals simultaneously in a direct marketing campaign. Among the N individuals, the proportion of the “respondents” who will eventually respond to the request is π. We call π the “ultimate response rate,” which is an unknown constant that should be estimated empirically.

Due to procrastination, even those respondents do not reply immediately. For each respondent, let p be the probability that he or she replies during a given day, and q = 1– p denote the daily “delay rate” of a respondent. Thus, the number of Bernoulli trials for each respondent to react is a geometric distribution with a parameter q.

Chun (2012) considered the geometric response model with the two parameters, π and q, in which the expected number of daily responses is decreasing over time, as shown in Figure 1.1(b). Now, we assume that each reply will be delivered d days later (0≤ d <∞), and the “delivery time” d is a discrete random variable. At the cost of introducing the additional variable d, we can represent various types of response patterns with different locations and shapes. Figure 1.2 illustrates the flowchart of responses during the first three days.

Figure 1.2

Figure 1.2 Flowchart of response patterns during the first three days.

For a respondent, let Pi be the probability that the reply of a respondent will be received i days after the launch of a direct marketing campaign. As shown in Figure 1.2, Pi does not depend on π, but it is a function of the unknown q and d . (Various types of functional forms of Pi will be considered in the next section.) The probability of receiving a series of responses, y={y1, y2, ..., yk}, during the first k days can be described as a multinomial distribution with (k+1) classes:

Equation (1-4)

from which we can find the expected values of yi and si as follows:

Equation (1-5)

01equ05.jpg

Equation (1-6)

01equ06.jpg

If we have the estimates of the parameters π, q, and d, we can predict the expected number of responses by a certain time and anticipate the time period needed to achieve a certain level of responses. Thus, our primary goal is to estimate π, q, and d empirically based on the sample observations y={y1, y2, ..., yk}.

Suppose that response data y={y1, y2, ..., yk} is available at time k. It follows from the multinomial distribution in (1-5) that the “likelihood function” of π is

Equation (1-7)

01equ07.jpg

The maximum likelihood estimator of ∝ maximizes this likelihood function in (1-7). It is well known that the optimal value that maximizes the likelihood function Ly(π) also maximizes its log-likelihood function, ln Ly(π). Therefore, it is more convenient to find the maximum likelihood estimator of π from the following log-likelihood function:

Equation (1-8)

If we take the first-order derivative with respect to π and set the equation equal to 0, we have

Equation (1-9)

01equ09.jpg

Solving this equation gives us the maximum likelihood estimator of the response rate π, as follows:

Equation (1-10)

01equ10.jpg

If we plug common.jpg in (1-10) into the log-likelihood function in (1-8) and rearrange the expression, we have

Equation (1-11)

01equ11.jpg

where ∝ denotes “is proportional to.”

The maximum likelihood estimates common1.jpg and common2.jpg are the ones that maximize this log-likelihood function in (1-11). Any optimization software, such as Microsoft Excel Solver, can be used to find the maximum likelihood estimates of q and d. With common1.jpg and common2.jpg, we then find the maximum likelihood estimate of π from (1-10).

Note that Pi is a function of q and d, where the delay rate q is an unknown constant, and the delivery time d is a random variable. If a specific distribution of the delivery time d is given, then we can specify the probability Pi in the log-likelihood function in (1-11). In the next section, we consider three different types of probability distribution function of the delivery time d.

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