Cumulative Incidence function and competing risks Let $R_1,...,R_K$ be competing risks and put $T=\min(R_1,...,R_K)$ and $\delta = i$ if $T=R_i$. The cumulative incidence function is then defined as $F_i(t):=P(T\le t,\delta = i)$. The cause-specific hazard rate is then defined as $$h_i(t)=\lim_{\Delta t\to 0}\frac{P(t\le T<t+\Delta t,\delta = i|T\ge t)}{\Delta t}.$$It is then stated that $$F_i(t)=\int_0^th_i(u)\exp(-H_T(u))du,$$where $H_T(u)=\sum_{j=1}^K\int_0^th_j(u)du$ (the cumulative hazard rate of $T$). However I can not see how this follows and could use some help on how to show it. If you need more information let me know. Thanks!
 A: Wikipedia has some good pages on the relationships between the event-time CDF, survival function, and cumulative hazard.
$h(t)$ can be seen as $f(t)/S(t)$, the pdf divided by the the survival function.  Additionally,  $\text{exp}[-H(t)]=S(t)$.  It should then be clear that
$$\int_0^t h(u)\cdot \text{exp}[-H(u)] du=\int_0^t \frac{f(u)}{S(u)}S(u) du=\int_0^t f(u) du=F(t). $$
For cause-specific risks it appears we are not conditioning on a risk but incorporating the risk type within the probability statement, i.e. $h_i(t)=f_i(t)/S(t)=f(t,\delta)/S(t)=h(t,\delta)$.  Then
\begin{eqnarray}
\int_0^t h_i(u)\cdot \text{exp}[-H_T(u)] du&=&\int_0^t h(u,\delta)\cdot \text{exp}[-H_T(u)] du\\
&=&\int_0^t \frac{f(u,\delta)}{S(u)}S(u) du\\
&=&\int_0^t f(u,\delta) du\\
&=&\int_0^t f_i(u) du\\
&\overset{\text{by definition}}{=}&F_i(t).
\end{eqnarray}
Some of the steps are unnecessary but serve as a reminder that we aren't conditioning on a particular type of risk.  Often times a subscript can suggest a conditional probability.  Let me know if your question is regarding something else.
