### 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…

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…