Census approximation exposed to risk

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!

 

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.

 

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

 

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!