CBD Family¶

Probability of death, not mortality rate

CBD-family models use the logit link and the Binomial distribution. fit.fitted_rates and fc.rates return the probability of death \(q_{xt}\), not the central mortality rate \(\mu_{xt}\). See Rates vs Probabilities for conversion formulas and guidance on exposure type.

CBD (Cairns-Blake-Dowd 2006)¶

\[\text{logit}(q_{xt}) = \kappa_t^{(1)} + (x - \bar{x})\,\kappa_t^{(2)}\]

fit = ps.cbd().fit(data.deaths, data.exposures,
                   ages=data.ages, years=data.years)

M6 (CBD + cohort)¶

\[\text{logit}(q_{xt}) = \kappa_t^{(1)} + (x-\bar{x})\,\kappa_t^{(2)} + \gamma_{t-x}\]

fit = ps.m6().fit(data.deaths, data.exposures,
                  ages=data.ages, years=data.years)

M7 (CBD + quadratic + cohort)¶

Adds a quadratic age effect:

\[\text{logit}(q_{xt}) = \kappa_t^{(1)} + (x-\bar{x})\,\kappa_t^{(2)} + [(x-\bar{x})^2 - \hat\sigma^2_x]\,\kappa_t^{(3)} + \gamma_{t-x}\]

fit = ps.m7().fit(data.deaths, data.exposures,
                  ages=data.ages, years=data.years)

M8 (CBD + age-at-zero cohort)¶

Uses a centered cohort age function with pivot age \(x_c\):

\[\text{logit}(q_{xt}) = \kappa_t^{(1)} + (x-\bar{x})\,\kappa_t^{(2)} + (x_c - x)\,\gamma_{t-x}\]

fit = ps.m8(xc=89.5).fit(data.deaths, data.exposures,
                          ages=data.ages, years=data.years)

References¶

Cairns, A.J.G., Blake, D. & Dowd, K. (2006). A Two-Factor Model for Stochastic Mortality with Parameter Uncertainty. Journal of Risk and Insurance, 73(4), 687–718.