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…

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Tags: Whittaker smoothing, splines, P-splines, penalty function

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