Makeham Log-Quadratic Mortality Law (GM(1,3)) - 1988 — makeham

The generalised Gompertz-Makeham graduation formula of Forfar, McCutcheon and Wilkie: a Makeham constant plus a Gompertz whose log hazard carries a quadratic term, \(\mu_x = A_0 + K \exp(B_1 x - B_2 x^2)\). The quadratic term bends the exponential rise downward, so the hazard decelerates at the oldest ages. This is the family UK pensioner tables are graduated with (the CMI S2 and 08 series) and the best-fitting law for the Canadian CPM2014 experience. SIGN IS A CHOICE: the published fits put a negative coefficient on the square, so it is written here as \(-B_2 x^2\) with \(B_2 > 0\), the branch the engine's positive parameters permit; the accelerating branch is out of reach.

Usage

makeham_logquad(x, par = NULL)

Arguments

x: vector of age at the beginning of the age classes.
par: parameters of the selected model. If NULL the default values are assigned automatically.

Value

A list of rates and model parameters.

Examples

makeham_logquad(x = 0:100)
#> $hx
#>   [1] 0.002000000 0.002104066 0.002216527 0.002337765 0.002468145 0.002608014
#>   [7] 0.002757689 0.002917457 0.003087569 0.003268230 0.003459603 0.003661793
#>  [13] 0.003874849 0.004098754 0.004333424 0.004578701 0.004834350 0.005100053
#>  [19] 0.005375409 0.005659928 0.005953032 0.006254054 0.006562235 0.006876727
#>  [25] 0.007196595 0.007520819 0.007848297 0.008177851 0.008508232 0.008838128
#>  [31] 0.009166170 0.009490942 0.009810992 0.010124837 0.010430980 0.010727919
#>  [37] 0.011014159 0.011288225 0.011548672 0.011794103 0.012023176 0.012234619
#>  [43] 0.012427240 0.012599941 0.012751725 0.012881707 0.012989125 0.013073343
#>  [49] 0.013133861 0.013170318 0.013182494 0.013170318 0.013133861 0.013073343
#>  [55] 0.012989125 0.012881707 0.012751725 0.012599941 0.012427240 0.012234619
#>  [61] 0.012023176 0.011794103 0.011548672 0.011288225 0.011014159 0.010727919
#>  [67] 0.010430980 0.010124837 0.009810992 0.009490942 0.009166170 0.008838128
#>  [73] 0.008508232 0.008177851 0.007848297 0.007520819 0.007196595 0.006876727
#>  [79] 0.006562235 0.006254054 0.005953032 0.005659928 0.005375409 0.005100053
#>  [85] 0.004834350 0.004578701 0.004333424 0.004098754 0.003874849 0.003661793
#>  [91] 0.003459603 0.003268230 0.003087569 0.002917457 0.002757689 0.002608014
#>  [97] 0.002468145 0.002337765 0.002216527 0.002104066 0.002000000
#> 
#> $par
#>    A0     K    B1    B2 
#> 0.001 0.001 0.100 0.001 
#>