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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