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

Introducing the Product Integral

(Feb 26, 2018)

Of all the actuary's standard formulae derived from the life table, none is more important in survival modelling than:

\[{}_tp_x = \exp\left(-\int_0^t\mu_{s+s}ds\right).\qquad(1)\]

Stephen covered the derivation of this in a previous blog, but I want to look more closely at the right-hand side of equation (1).  In particular, we can find an entirely different representation of \({}_tp_x\) as a product integral, which leads to many insights in survival models.

Recall how the integral in equation (1) is constructed.  Choose a partition of the interval \([0,t]\), that is some sequence \(\Delta_1,\Delta_2,\ldots,\Delta_n\) of non-overlapping sub-intervals that exactly cover the interval.  Define…

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Tags: survival models, survival probability, force of mortality, product integral

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