### 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…

### 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…