Gompertz-Makeham family ch7

[kalky]

\[ \mu_{x} = GM(r,s) = \alpha_{1} + \alpha_{2} t + \dots \]

where t is linear function of x.

However, \( t = c^x \) is not a linear function. So, where am I wrong?


N.B. I substitute \( t = c^x \) into equation to get:

\[ \mu_{x} = GM( r,s ) = \alpha_{2} c^x + \dots Gompertz \]
\[ \mu_{x} = GM( r,s ) = \alpha_{1} + \alpha_{2} c^x + \dots Makeham\]

 

[Julie Lewis]

We can write:

\[ Bc^x = e^{lnBc^x}=e^{lnB + xlnc}\]

This is of the form:

\[ e^{\beta_0 + \beta_1 x} \]

So it's a member of the GM family as it's of the form polynomial + exp(polynomial). Note that the first polynomial is 0.

 

[kalky]

Tks Julie!

 

[Nabil Janmohamed]

Hi,

Can someone help me calculate the constants in Makeham's Law in the attached?

Many thanks,

Nabil

 

[Mark Mitchell]

You need to set up the three simultaneous equations, based on the form of Makeham's Law on page 32 of the Tables and the values of mu_20, mu_30 and mu_40, and solve them.

I'm not going to do all the work for you, but to start you off you could:

subtract the equation for mu_20 from the equation for mu_30 (call the resulting equation *)
subtract the equation for mu_30 from the equation for mu_40 (call the resulting equation #)

These two equations should have no A in them. If you divide # by * you should eliminate B too, leaving an equation only in c, so you can find its value.

Is this question taken from somewhere in the ActEd materials? If so, the method of solution ought to be shown for you.

 

[Nabil Janmohamed]

Thanks Mark,

Yes, you are absolutely correct and I have managed to solve it. Unfortunately, the question is from another reference I use for my studies alongside with Acted materials and the solution was shown but not the methodology.