Welcome to the hotseat. We've prepared a guide if you'd like to read more about how it works.

Don't the assumptions of uniform distribution of deaths and linearity of the force of mortality contradict one another?

0 votes
126 views
asked Jun 14 in BUS 3018F - Models by 25b14c23 (340 points)

In tutorial 4 question 2 ii (attached below) we were told to estimate the force of morality and initial rate of mortality using the UDD assumption. So I calculate \(\hat{u_x}\) which is an estimate for \(u_{x+0.5}\). Now to get an estimate for \(q_x\) I can either use the relationship which we prove in question 1 OR I assume that \(u_{x+t}\) is linear between ages x and x+1 and solve using $$ q_x = 1 - exp(-{\int {u_{x+t}dt}}) = 1 - exp(-0.5*(u_{x}+u_{x+1}))  = 1 - exp(-u_{x+0.5}) = 1 - exp(- \hat{u_x} )$$ 

which is what I did in the tutorial test and was marked correctly, but the relationship proven in question 1 contradicts the way I did it. If you could shed some light on what I am missing I would really appreciate it. 

image


x

1 Answer

+1 vote
answered Jun 15 by joshua_wort (1,390 points)
selected Jun 15 by 25b14c23
 
Best answer

The UDD assumption does indeed contradict, in the strict sense, the assumption that \(\mu_{x+t}\) is linear over ages x to x+1 as can be seen by the answer given in question 1, $$\mu_{x+t} = \frac{q_x}{1 - t*q_x}$$

for \(0\le t \lt 1\).

However, over the small range from 0 to 1, this function can be considered approximately a linear function, so that is why the linear assumption for force of mortality may be used with the UDD assumption.

Below is the force of mortality function under UDD assumption where I have decided to set \(q_x = 0.4\) for argument's sake, and this can be seen to be approximately a linear function over the range from 0 to 1.

In question 2(ii). we are giving an estimate so we aren't too perturbed with considering \(\mu_{x+t}\) as a linear function.

It must be noted that the UDD assumption and the constant force of mortality assumption definitely contradict each other and should never be used together.

image   

commented Jun 15 by 25b14c23 (340 points)

Thanks for the answer, it makes a lot more sense now. However, I'm confused as to why we even make the linearity of \(u_{x+t}\) assumption if we already have this nice relationship that is more precise and easier to use. Like I get that it is an estimate, but why further muddle up the estimate with more assumptions when the UDD assumption is more than enough? 

The estimate for \(u_{x}\) will be slightly different when making the linearity assumption and the UDD assumption compared to just making the UDD assumption, but we can get this better more precise estimate with less work and less assumptions, so I would think that we should just make the UDD assumption, right?


Thanks

commented Jun 15 by joshua_wort (1,390 points)

Yes, you are correct, it definitely would be better to just assume the uniform distribution of deaths assumption.

...