### Don't cut corners

#### (Oct 7, 2014)

An important class of mortality-projection models is the Cairns-Blake-Dowd (CBD) family. These models are described in a landmark paper by Cairns et al (2009).  Three of the most important of the CBD models are M5, M6 and M7, as defined below for age $$x$$ and calendar year $$y$$:

 M5 $$\log \mu_{x,y} = \kappa_{0,y} + \kappa_{1,y}S(x)$$ M6 $$\log \mu_{x,y} = \kappa_{0,y} + \kappa_{1,y}S(x) + \gamma_{y-x}$$ M7 $$\log \mu_{x,y} = \kappa_{0,y} + \kappa_{1,y}S(x) + \gamma_{y-x} + \kappa_{2,y}Q(x)$$

where:

\eqalign{S(x) &= \left(x - \bar x\right)\\ Q(x) &= \left(x - \bar x\right)^2-\hat\sigma^2\\ \hat\sigma^2 &= \displaystyle\frac{1}{n_x}\sum_{i=1}^{n_x} (x_i-\bar x)^2}

The original…