Промышленный лизинг Промышленный лизинг  Методички 

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The second peak in the M is coincident with the end of the initial promotion that offers introductory pricing. This promo typically lasts for about 3 months, and then customers have to start paying full price. Many decide that they no longer really want the service. It is quite possible that many of these customers reappear to take advantage of other promotions, an interesting fact not germane to this discussion on hazards but relevant to the business.

After the first 3 months, the hazard function has no more really high peaks. There is a small cycle of peaks, about every 4 or 5 weeks. This corresponds to the monthly billing cycle. Customers are more likely to stop just after they receive a bill.

The chart also shows that there is a gentle decline in the hazard rate. This decline is a good thing, since it means that the longer a customers stays around, the less likely the customer is to leave. Another way of saying this is that customers are becoming more loyal the longer they stay with the company.

Censoring

So far, this introduction to hazards has glossed over one of the most important concepts in survival analysis: censoring. Remember the definition of a hazard probability, the number of stops at a given time t divided by the population at that time. Clearly, if a customer has stopped before time t, then that customer is not included in the population count. This is most basic example of censoring. Customers who have stopped are not included in calculations after they stop.

There is another example of censoring, although it is a bit subtler. Consider customers whose tenure is t but who are currently active. These customers are not included in the population for the hazard for tenure t, because the customers might still stop before t+1-here today, gone tomorrow. These customers have been dropped out of the calculation for that particular hazard, although they are included in calculations of hazards for smaller values of t. Censoring-dropping some customers from some of the hazard calculations-proves to be a very powerful technique, important to much of survival analysis.

Lets look at this with a picture. Figure 12.7 shows a set of customers and what happens at the beginning and end of their relationship. In particular, the end is shown with a small circle that is either open or closed. When the circle is open, the customer has already left and their exact tenure is known since the stop date is known.

A closed circle means that the customer has survived to the analysis date, so the stop date is not yet known. This customer-or in particular, this customers tenure-is censored. The tenure is at least the current tenure, but most likely larger. How much larger is unknown, because that customers exact stop date has not yet happened.



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time

Figure 12.7 In this group of customers who all start at different times, some customers are censored because they are still active.

Lets walk through the hazard calculation for these customers, paying particular attention to the role of censoring. When looking at customer data for hazard calculations, both the tenure and the censoring flag are needed. For the customers in Figure 12.7, Table 12.2 shows this data.

It is instructive to see what is happening during each time period. At any point in time, a customer might be in one of three states: ACTIVE, meaning that the relationship is still ongoing; STOPPED, meaning that the customer stopped during that time interval; or CENSORED, meaning that the customer is not included in the calculation. Table 12.3 shows what happens to the customers during each time period.

Table 12.2 Tenure Data for Several Customers

CUSTOMER

CENSORED

TENURE



Table 12.3 Tracking Customers over Several Time Periods

CUSTOMER

CENSORED

LIFETIME

TIME 0

TIME 1

TIME 2

TIME 3

TIME 4

TIME 5

ACTIVE

ACTIVE

ACTIVE

ACTIVE

ACTIVE

ACTIVE

ACTIVE

ACTIVE

ACTIVE

ACTIVE

STOPPED

CENSORED

ACTIVE

ACTIVE

ACTIVE

STOPPED

CENSORED

CENSORED

ACTIVE

ACTIVE

ACTIVE

ACTIVE

CENSORED

CENSORED

ACTIVE

ACTIVE

STOPPED

CENSORED

CENSORED

CENSORED

ACTIVE

ACTIVE

CENSORED

CENSORED

CENSORED

CENSORED

ACTIVE

STOPPED

CENSORED

CENSORED

CENSORED

CENSORED



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