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Adjusting census data - exposed to risk - question 6 tut 3

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asked May 29 in BUS 3018F - Models by anonymous

When adjusting our census data to get the correct age label is it correct to look at the population at different ages at time t? Or should we always look at the population at different times and decide which applies to our age label? 

For example, in question 6 we were given census data age x nearest (Px(t)) and needed age x last (Px'(t)) to match the death data. 

Consider t = 01/01/05 only. We could look at data for the population aged x at different times and would arrive at the answer Px'(01/01/05) = 1/2 (Px(01/01/05)) or we could consider different ages at a fixed t - Px'(t) = 1/2 Px(t) + 1/2 Px+1(t) as assuming UBD, 1/2 of those aged x+1 nearest at time t would have been age x last at time t. Is my thinking here completely incorrect?

1 Answer

+1 vote
answered May 30 by Murray (660 points)
It always depends on how you are adjusting your census data.
So, for this question we will see both methods used to get the final answer. First we are estimating exposure so we need to find:
$$E_x^c=\int_{0}^{2}P_x^{last}(t)dt$$
Here we assume that exposure varies linearly between census dates. This is so we can evaluate the integral and is the reason we use different t's in the next line. It follows that:
$$E_x^c=\frac{1}{2}(P_x^{last}(01/01/05)+P_x^{last}(01/01/06))+\frac{1}{2}(P_x^{last}(01/01/06)+P_x^{last}(01/01/07)$$
But we don't have Px' so we need to approximate it using an assumption and our Px's. You correctly showed using the assumption of UBD over the calendar year that:
$$P_x^{last} = \frac{1}{2}(P_x^{nearest}(t)+P_{x+1}^{nearest}(t))$$
This is because, under the assumption, half the lives for Px' will be aged x under Px and the other half will be Px+1.
The answer follows by changing all the Px' to Px and Px+1's.
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