# Estimating the rate of mortality using initial exposure or using force of mortality already calculated

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I refer to Tutorial 5 part 2:

You are carrying out an investigation into the mortality experience of pensioners. You have been given the following data:

• the number of deaths aged x nearest birthday at death
• the years of exposure to risk of lives aged x next birthday during the period of investigation (derived directly from dates of birth and exposure)
Question i) asks us to identify the most appropriate rate interval and then to estimate the force of mortality for lives aged x exactly.

Question ii) then asks us to explain how to estimate $$q_{x}$$.
Since in question i) I manipulated the central exposed to risk data to find an appropriate $$E_{x}^{c}$$ such that the principle of correspondence holds, my next assumption was that deaths were distributed uniformly over the rate interval, so I can estimate $$E_{x}$$ by $$E_{x}=E_{x}^{c}+\frac{1}{2}d_{x}$$
Why would this be incorrect, with the correct answer to use the force of mortality that we calculated in i) to arrive at $$q_{x}=1-e^{-\frac{1}{2}(\mu_{x}+\mu_{x+1})}$$?

+1 vote
answered Jun 17, 2017 by (1,490 points)
selected Jun 17, 2017

It would be incorrect in using $$E_x$$ since you would then be finding $$q_{x-\frac{1}{2}}$$, as the start of the rate interval is for lives aged $$x-\frac{1}{2}$$ exact. Instead you are asked to estimate $$q_x$$, so that is why you use the assumption that the force of mortality is linear between ages $$x$$ and $$x+1$$ in order to find $$q_x = 1 - e^{-\frac{1}{2}(\mu_x + \mu_{x+1})}$$