Smooth Models Meet Lumpy Data

(Aug 15, 2018)

Most of the survival models used by actuaries are smooth or piecewise smooth; think of a Gompertz model for the hazard rate, or constant hazard rates at individual ages.  When we need a cumulative quantity, we use an integral, as in the cumulative hazard function, \(\Lambda_x(t)\):

\[ \Lambda_x(t) = \int_0^t \mu_{x+s} \, ds. \qquad (1) \]

Mortality data, on the other hand, are nearly always lumpy.  A finite number of people, \(d_{x+t_i}\) say, die at a discrete time \(t_i\), one of a set of observed times of death \(t_1, t_2, \ldots, t_r\).  Then when we need a cumulative quantity, we use a sum.  We saw in a previous blog that if \(l_{x+t_i^-}\) was the number of persons being observed just before time \(t_i\), then…

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Tags: Nelson-Aalen

The Karma of Kaplan-Meier

(May 7, 2018)

Our new book, Modelling Mortality with Actuarial Applications, describes several non-parametric estimators of two quantities:

  1. The survival function, \(S_x(t)\), defined as the probability that a person now aged \(x\) will survive at least \(t\) years (\({}_tp_x\) to actuaries), and
  2. The integrated hazard function, \(\Lambda_x(t) = \displaystyle\int_0^t\mu_{x+s}ds\).

The estimators of the above quantities are based on two items of data collected at the times of the observed deaths (denoted by \(t_1,t_2,\ldots,t_n\)):

  1. The number, \(d_{x+t_i}\), who died at time \(t_i\), and
  2. The number, \(l_{x+t_i^-}\), who were alive and under observation immediately before time \(t_i\) (which time we denote…

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Tags: Kaplan-Meier, Nelson-Aalen, Fleming-Harrington, product integral

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