Gompertz distribution. Motivation.
A survival function can be expressed as:
The transformation is constrained so that all 3 of the following statements hold:
From here on, we are concerned with the situation where .
Gompertz (1825) assumed that took the form:
Gompertz describes the motivation in detail, relating it to a geometric progression of deaths within long fixed length periods. The transformation satisfies the constraints above, on the condition that
. That must be positive is implied by
the survival function being positive function.
Gompertz distribution. Properties.
Below are some properties of the Gompertz distribution.
Mode
Differentiating and equating to zero, we find the mode:
If , then the mode is at . If ,
then the mode is at .
Wikipedia
also notes that when the mode is positive, the cumulative distribution evaluated
at is always between 0 and 0.6321:
Translation
If is Gompertz distributed with parameters
, then a forward shifted variable is
Gompertz distributed with parameters .
Proof.
With a change of variable, , we will show that
where is the same
survival function as , but with the parameter being set to .
With this done, it will follow that \( S(t|t>v) = S*(t').
Let .
The consequence of this truncating a Gompertz distribution at time
(i.e. setting to zero the all probability mass at times less than )
and making a variable change such that leaves a Gompertz
distribution with parameters .
and , definitions
Let be the codomain of a random variable.
Typically, the codomain is non-negative, .
Use to denote a generic value in the codomain. Let
denote the infimum of the codomain (typically ).
Let be a probability density function of the
underlying random variable. We then make a number of definitions:
- Cumulative distribution function: .
- Survival function: .
- Hazard function: .
Survival function. Intuition.
The survival function maps a time to a probability mass,
representing the probability that the event has not occurred yet, by
time . This is the most natural representation to work with
when answering the question: what is the probability I will live until at
least age 46.
The survival function acts as a re-normalizing factor in
that allows to be transformed with the information that the event has
not occurred by time . While is normalized by , should be normalized by the remaining probability mass
, which will be less that 1.
When is the hazard function:
and so, the hazard function is the continually re-normalized density
function. When worrying about dying, the hazard function tells
us the danger of dying at , assuming that one has lived until
. If you knew that you would go to war for 4 years once you reach
18, then your hazard function would sharply spike at 18 and then sharply
drop again when the war ends, or you get discharged. High hazard values
denote times at which you should exercise caution.