The Makeham hazard with the exponential term in mode/dispersion form, \(\mu_x = (1/\sigma) \exp((x - M)/\sigma) + C\).
Examples
makeham0(x = 45:90)
#> $hx
#> [1] 0.07828767 0.08901739 0.10123670 0.11515240 0.13099999 0.14904768
#> [7] 0.16960090 0.19300749 0.21966358 0.25002029 0.28459137 0.32396190
#> [13] 0.36879818 0.41985901 0.47800852 0.54423084 0.61964670 0.70553242
#> [19] 0.80334151 0.91472928 1.04158083 1.18604298 1.35056057 1.53791788
#> [25] 1.75128569 1.99427502 2.27099815 2.58613831 2.94502887 3.35374362
#> [31] 3.81919958 4.34927403 4.95293786 5.64040735 6.42331712 7.31491699
#> [37] 8.33029622 9.48663890 10.80351475 12.30321035 14.01110625 15.95610657
#> [43] 18.17112813 20.69365753 23.56638560 26.83792985
#>
#> $par
#> sigma M C
#> 7.692308 49.000000 0.001000
#>
#> $Sx
#> [1] 5.284493e-01 4.860981e-01 4.420468e-01 3.967760e-01 3.508881e-01
#> [6] 3.050998e-01 2.602226e-01 2.171279e-01 1.766976e-01 1.397600e-01
#> [11] 1.070178e-01 7.897608e-02 5.588246e-02 3.769302e-02 2.407493e-02
#> [16] 1.445107e-02 8.082058e-03 4.170296e-03 1.963266e-03 8.325853e-04
#> [21] 3.134859e-04 1.030802e-04 2.904909e-05 6.867341e-06 1.329079e-06
#> [26] 2.048120e-07 2.434808e-08 2.153938e-09 1.360954e-10 5.861448e-12
#> [31] 1.631584e-13 2.762747e-15 2.656017e-17 1.340135e-19 3.245089e-22
#> [36] 3.405655e-25 1.379309e-28 1.888871e-32 7.523940e-37 7.344083e-42
#> [41] 1.445072e-47 4.589235e-54 1.826097e-61 6.823507e-70 1.724143e-79
#> [46] 2.026764e-90
#>