Exposed to risk

Can anyone tell me the basic difference between initial exposed to risk and central exposed to risk? I mean to ask if we say that tha IETR for a person is 11months and CETR is 7months then what actually mean by this two numbers.........Please do reply as early as possible.

Thanks in advance!!

 

They're just two different ways of measuring the length of time a person is exposed to the risk of dying.

If Initial E = 11 months, then Central E should be 11-6months = 5.

What I would like to know is, why do we use Initial E in the binomial model and Central E in Poisson model?

 

ActEd's "lightbulb boxes" in Chapter 10 prove quite helpful here...

Suppose you observe a life over the age interval (x+a, x+b), with 0 < a < b < 1.

Suppose the life survives to age x+b. You've observed the life for (b-a) years. It contributes (b-a) to both the initial and central exposed to risk.

Now suppose the life dies at age x+t, with a < t < b. You've observed the life for (t-a) years.

• It contributes (t-a) to the central exposed to risk...
• but it contributes (1-a) to the initial exposed to risk.

So the central exposed to risk is the actual time you spend observing lives. The initial exposed to risk has some sort of "adjustment" in respect of those lives observed to die.

 

If I remember correctly, the reason we use two different Exposed to Risk is because each is suited to a different model. The exposed to risk is determined by what is appropriate for a given model.

Poisson model uses the central exposed to risk.
It is a continuous distribution with the "probabilities" (read force of mortality) apply at each INSTANT to the exposed lives at that instant.
Decrease the number of lives (say if some die) then the exposed lives reduces. If you think back to the derivation of formulae for the poisson process, you always consider loads of instants in time and add them up.
So we need the "exact time" to calculate the exposure.

You can think of the exposure being an average for the period, which may be a bit like the exposure at the middle of the period, hence the term central exposed to risk.

Binomial model looks at number of exposed lives at the START and counts how many are alive at end of period. It is discrete, either you die or not in the period, and doesn't matter if it's near the start or the end of the period.
So an INITIAL exposed to risk is used.

 

Thanks didster

I understand the binomial one, but not Poisson. I thought poisson is a discrete process? If it can be approximately Normal, then the same can be said for the Binomial model.

 

I'll try to be a bit more specific, although it's difficult since it's been a while.

The survival model is the Poisson process, which is not to be confused with the Poisson distribution.

Poission process is a discrete state, continuous time model.
Here we could have the states being the number of lives present.

The Poission distribution is the distribution of a random variable, being the number of states that you move. The number of states is discrete, and Poisson distribution is discrete random variable.

So the movement is continuous (in time), but the counting of the movements is discrete. That's the link between the two "Poisson"

The other link with the Poisson process is that the times between jumps is an exponential distribution (and each time is independant).

I'm sure there is something in the notes about the poisson process. Have a read of that (including the maths involved), then try to apply such a model to lives, thinking about how to measure the Exposed to Risk.

 

Thanks didster.

I did read about Poisson process, but I failed to see the link between that and the whole survival model. Maybe that's why I didn't get it.

Will read it again and try to see the big picture.