I'm trying to get intuition for each of the main functions in actuarial science (specifically for the Cox Proportional Hazards Model). Here's what I have so far:
- $f(x)$: starting at the start time, the probability distribution of when you will die.
- $F(x)$: just the cumulative distribution. At time $T$, what % of the population will be dead?
- $S(x)$: $1-F(x)$. At time $T$, what % of the population will be alive?
- $h(x)$: hazard function. At a given time $T$, of the people still alive, this can be used to estimate how many people will die in the next time interval, or if interval->0, 'instantaneous' death probability.
- $H(x)$: cumulative hazard. No idea.
What's the idea behind combining hazard values, especially when they are continuous? If we use a discrete example with death rates across four seasons, and the hazard function is as follows:
- Starting at Spring, everyone is alive, and 20% will die
- Now in Summer, of those remaining, 50% will die
- Now in Fall, of those remaining, 75% will die
- Final season is Winter. Of those remaining, 100% will die
Then the cumulative hazard is 20%, 70%, 145%, 245%?? What does that mean, and why is this useful?