# BUS3018F: reconciling the mathematical specification of the initial exposed to risk with its definition

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Hi all!

Let's get a short discussion going on the initial exposed to risk. Recall that $$E_{x}$$ arises as a mathematical construct in the generalized binomial model (you don't have to worry about the derivation), and so the mathematical definition is "the summation of the central exposed to risk (which is a period of time) and (1-$$t_{i}$$)- which is also a period of time - for all the deaths".

On the other hand, the qualitative definition of $$E_{x}$$ is "the equivalent (or I would say expected) number of lives which would have to be alive at the beginning of the rate interval (e.g. exact age $$x$$)  in order to have given rise to the number of deaths observed, had there been no entrants or exits (other than death).

Now, how can we reconcile the mathematical definition of $$E_{x}$$ with the qualitative definition? Also, how can $$E_{x}$$ approximate a number of lives when it is only a summation of time?

I am interested to see YOUR responses (including tutors or others not involved in the course) please. Give it a bash!

+1 vote
answered Apr 3, 2018 by (260 points)

I think that we can begin by assessing a particular use of the initial exposed to risk, which is to derive an estimate for the rate of mortality under the binomial (simple) model. In this case we need to decide (based on our assumptions) what information from our investigation will be necessary. Prospectively we need to know the number of lives (all aged x at the beginning of our investigation) that will be under observation, as well as the number of deaths (of people who were aged x at start of observation) that occurred during that year of investigation. With this, we can estimate a rate of mortality for people aged x. Considering that these binomial assumptions barely hold in practice, the model needs to be generalized. Without trying to unpack the Balducci assumption, it can be shown that a relationship between the central exposed to risk (which is far more intuitive to calculate) can be used to estimate the initial exposed to risk. That is, mathematically we can create the right conditions needed to use the binomial model without adhering to its assumptions. In this latter case our exposed to risk is duration based. The way I understand it is that it is also indirectly duration based in the simple binomial model. The connection arises when we consider the fact that by summing the duration of all people's lives in a given year, and adding back the duration of  non-life (that is the total remainder of the year that those who died didn't experience) then the result you get is equal to the  number of lives you expected to observe throughout the year.

Note: I deliberately left out any effects of censoring or truncation.

commented Apr 5, 2018 by (1,350 points)

Good answer! I like the fact that you immediately linked my question to the USE for the initial exposed to risk.

Also, remember that the initial exposed to risk can be viewed in “part persons”, and need not be an integer.