Combining proportions dying as you do is not giving you cumulative hazard. Hazard rate in continuous time is a conditional probability that during a very short interval an event will happen:
$$h(t) = \lim_{\Delta t \rightarrow 0} \frac {P(t<T \le t + \Delta t | T >t)} {\Delta t}$$
Cumulative hazard is integrating (instantaneous) hazard rate over ages/time. It's like summing up probabilities, but since $\Delta t$ is very small, these probabilities are also small numbers (e.g. hazard rate of dying may be around 0.004 at ages around 30). Hazard rate is conditional on not having experienced the event before $t$, so for a population it may sum over 1.
You may look up some human mortality life table, although this is a discrete time formulation, and try to accumulate $m_x$.
If you use R, here's a little example of approximating these functions from number of deaths at each 1-year age interval:
dx <- c(3184L, 268L, 145L, 81L, 64L, 81L, 101L, 50L, 72L, 76L, 50L,
62L, 65L, 95L, 86L, 120L, 86L, 110L, 144L, 147L, 206L, 244L,
175L, 227L, 182L, 227L, 205L, 196L, 202L, 154L, 218L, 279L, 193L,
223L, 227L, 300L, 226L, 256L, 259L, 282L, 303L, 373L, 412L, 297L,
436L, 402L, 356L, 485L, 495L, 597L, 645L, 535L, 646L, 851L, 689L,
823L, 927L, 878L, 1036L, 1070L, 971L, 1225L, 1298L, 1539L, 1544L,
1673L, 1700L, 1909L, 2253L, 2388L, 2578L, 2353L, 2824L, 2909L,
2994L, 2970L, 2929L, 3401L, 3267L, 3411L, 3532L, 3090L, 3163L,
3060L, 2870L, 2650L, 2405L, 2143L, 1872L, 1601L, 1340L, 1095L,
872L, 677L, 512L, 376L, 268L, 186L, 125L, 81L, 51L, 31L, 18L,
11L, 6L, 3L, 2L)
x <- 0:(length(dx)-1) # age vector
plot((dx/sum(dx))/(1-cumsum(dx/sum(dx))), t="l", xlab="age", ylab="h(t)",
main="h(t)", log="y")
plot(cumsum((dx/sum(dx))/(1-cumsum(dx/sum(dx)))), t="l", xlab="age", ylab="H(t)",
main="H(t)")
Hope this helps.
Best Answer
All of these terms are standard in actuarial science and all of them apply to all distributions (but when I have seen these terms in studying for exams, we're almost always talking about distributions that are defined only for nonnegative reals). $H(t)$ is the cumulative hazard function, and for any distribution is defined as $$H(t) = \int_0^t h(x) \,dx.$$ Notice the name makes perfect sense with this definition, since we are "adding" up the hazard function up to a certain point to get the cumulative hazard function. Now, since $$f(t) = F'(t) = -S'(t)$$ then we have $$h(t) = \frac{f(t)}{S(t)} = \frac{-S'(t)}{S(t)} = -\frac{d}{dt} (\ln S(t)).$$ Finally, that means we have $$H(t) = \int_0^t -\frac{d}{dx} (\ln S(x)) \,dx = -\ln S(t)$$ since $S(0)$ is usually required to be 1 and thus $\ln S(0) = 0$.