Mortality Estimation and Graduation
This domain tests how crude mortality rates are estimated from exposed-to-risk data and how those rates are smoothed into a graduated table suitable for pricing and valuation.
Exposed to risk and crude rates
- The central exposed to risk E_x^c is the total time lived in the age interval, often approximated by the mid-year population. Under a constant force, q_x = 1 - exp(-d_x/E_x^c).
- The initial exposed to risk E_x^i counts lives at the start of the interval. Then q_x is estimated directly as d_x/E_x^i.
- Census approximations match census dates: for ages x to x+1, a mid-year census approximates the central exposure; a census at the start of the year approximates initial exposure.
- Graduation fits a smooth curve to crude rates. Parametric laws include Gompertz (mu_x = Bc^x), Makeham (mu_x = A + Bc^x) and Thiele. Non-parametric methods include splines and Whittaker-Henderson.
Standard tables and tests
A standard table gives expected deaths E_x q_x^s. The standardised mortality ratio is SMR = observed deaths / expected deaths. Graduation is checked with chi-square tests, signs tests, runs tests, cumulative deviation tests and serial correlation tests.
Worked example
For age 40, initial exposure is 50,000 and deaths are 400, so crude q_40 = 400/50,000 = 0.008. If the central exposure is instead 49,600, the constant-force estimate is q_40 = 1 - exp(-400/49,600) = 0.00803. If a standard table predicts 360 deaths, SMR = 400/360 = 1.111, indicating 11.1% more mortality than standard.
Traps that cost marks
Do not divide deaths by the wrong exposure: central exposure gives a force-based estimate, while initial exposure gives a direct q_x estimate. Do not count a death in the wrong age interval, and do not apply a chi-square test when expected deaths are too small. Graduation must preserve total expected deaths if the method is required to be unbiased.