### Back to the future with Whittaker smoothing

#### (Aug 7, 2015)

Many actuaries will be familiar with Whittaker smoothing (1923) but few will be aware of the close connection between this early method and the method of P-splines. The purpose of this blog is to explain this connection.

Figure 1 is a typical plot of log(mortality) of the kind of data we might want to smooth; here we use UK male data for 2011 for ages 2 to 30 taken from the Human Mortality Database.

Figure 1: Crude and Whittaker smoothed log mortality rates for UK males ages 2 to 30 in 2011.

Let $$y_i = \log(d_i/e_i), i = 1, \ldots, n$$, be our data where the $$d_i$$ are the observed deaths at age $$i$$ and the $$e_i$$ are the corresponding central exposures, and let $$\mu_i, i = 1, \ldots, n$$, be the candidate smooth values…