hacky M estimation from single-age data

M_spline.R

devtools::load_all()

# library(HMDHFDplus)
# mx <- readHMDweb("USA","mltper_1x1", 
#            username = Sys.getenv("us"), 
#            password = Sys.getenv("pw")) |> 
#   filter(Year == 2000) |> 
#   pull(mx)

mx <- c(0.00791, 0.00056, 0.00038, 0.00029, 0.00022, 0.00021, 0.00019, 
        0.00017, 0.00016, 0.00016, 0.00018, 0.00021, 0.00022, 0.00026, 
        0.00035, 0.00049, 0.00072, 0.00093, 0.00116, 0.00132, 0.00135, 
        0.00142, 0.00137, 0.00134, 0.00135, 0.00127, 0.00127, 0.0013, 
        0.00129, 0.00134, 0.00129, 0.0014, 0.00145, 0.00156, 0.00169, 
        0.00174, 0.0019, 0.00204, 0.00226, 0.00235, 0.0026, 0.00278, 
        0.003, 0.00333, 0.00359, 0.00384, 0.00419, 0.00455, 0.00502, 
        0.00523, 0.00554, 0.00588, 0.00643, 0.00683, 0.00735, 0.00825, 
        0.00901, 0.00981, 0.01076, 0.01177, 0.01267, 0.01369, 0.01531, 
        0.01674, 0.01841, 0.01974, 0.02148, 0.02364, 0.02566, 0.02785, 
        0.03064, 0.03332, 0.03745, 0.04012, 0.04332, 0.04772, 0.05175, 
        0.05643, 0.06245, 0.06951, 0.08043, 0.08598, 0.09718, 0.10638, 
        0.11855, 0.13146, 0.14549, 0.16027, 0.17495, 0.19596, 0.2124, 
        0.23314, 0.25763, 0.28428, 0.30846, 0.32839, 0.35464, 0.3818, 
        0.40972, 0.43823, 0.46716, 0.49631, 0.52548, 0.55449, 0.58312, 
        0.61121, 0.63857, 0.66507, 0.69056, 0.71495, 0.73814)
delta <- .01
lower <- 50
mx2   <- splinefun(mx ~ .5:110.5)(seq(lower,110,by=delta))
dx    <- -diff(c(1,exp(-cumsum(mx2*delta)),0))
# Mode
which.max(dx) * delta + lower