Census approximation exposed to risk

[Hells_angel]

For tutorial question (q5 from handout 3/4 or q15 from handout 2/2) part b you are asked to approximate mu wrt theta_3. I have outlined my understanding below but don't get the same answer as in the solutions, can you help?

theta_3 = # deaths aged x nearest birthday at previous 1 Jan

census data P'y(2001) = # lives aged x last birthday at 1 Jan 2001
Px(0) = # lives aged x nearest birthday at previous 1 Jan

if census life dies, last birthday on average 6 months prior
(estimating d_x and mu_x+1/2)

if theta_3 life dies, previous 1 Jan on average 6 months before and nearest birthday on average now
(estimating d_x and mu_x+1/2)

so P'x(2001)=Px(2001). For exposed to risk have to adjust since death data changes age on 1 Jan = census date.

So central EtR ~= 1/2 [ P'x(2001) + P'x+1 (2002) ]

= 1/2 [ Px(2001) + Px+1(2002) ]


Many thanks!

 

[vgvu99]

I didn't go to that tutorial but I believe the question is how to express the "# of lives aged x last birthday on 1 Jan" in terms of "# of lives aged x nearest birthday @ the 1 Jan prior to death"

Let Px'(t) = census data of those aged x nearest birthday on 1 Jan immediately before the date of the census.

So Central EtR ~= 1/2 [ Px'(0) + Px+1'(1) ]

A life aged x nearest birthday has the age range of [x-1/2, x+1/2] ==> The age last birthday of this life is either x -1 or x

Assuming that birthdays are uniformly distributed over the calendar year
==> Px'(0) = 1/2[Px-1(0) + Px(0)]

Similarly,
Px+1'(1) = 1/2[Px(1) + Px+1(1)]

Therefore you can work out the central exposed to risk correspondingly. I hope this is correct.

 

[vgvu99]

Btw, this question is very similar to Question 11.10 (page 28 - Chapter 11 - Exposed to Risk)

 

[Hells_angel]

Hey, thanks for that, same answer as the solutions. Sometimes get bit confused keeping the thought process logical so that helps. Will try the other question tonight,

Cheers!