- Abstract
- Introduction
- Preliminaries
- Delivery Time
- Customer Response Model
- Delivery Time Models
- Numerical Example
- Concluding Remarks
- References
- About the Authors

## 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** Flowchart of response patterns during the first three days.

For a respondent, let *P*_{i} 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, *P*_{i} does not depend on π, but it is a function of the unknown *q* and *d* . (Various types of functional forms of *P*_{i} will be considered in the next section.) The probability of receiving a series of responses, **y**={*y*_{1}, *y*_{2}, ..., *y*_{k}}, 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 *y*_{i} and *s*_{i} as follows:

#### Equation (1-5)

#### Equation (1-6)

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**={*y*_{1}, *y*_{2}, ..., *y*_{k}}.

Suppose that response data **y**={*y*_{1}, *y*_{2}, ..., *y*_{k}} is available at time *k*. It follows from the multinomial distribution in (1-5) that the “likelihood function” of π is

#### Equation (1-7)

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 *L*_{y}(π) also maximizes its log-likelihood function, ln *L*_{y}(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)

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

#### Equation (1-10)

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

#### Equation (1-11)

where ∝ denotes “is proportional to.”

The maximum likelihood estimates and 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 and , we then find the maximum likelihood estimate of π from (1-10).

Note that *P*_{i} 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 *P*_{i} 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*.