Competing-risk data, cause-specific hazard measures, and cumulative incidence

Fix a finite number $K\ge 1$ of causes on a probability space. Let $T_1,\dots ,T_K$ be measurable event times with values in $[0,\mathrm{\infty })$, and assume that the minimum is attained by a unique cause almost surely. The observed first failure time and its cause are $T:={\min }_{1\le j\le K}{T}_j,D:=\min \{j:T_j=T\}\in \{1,\dots ,K\},$ so $D$ is the smallest label attaining the minimum. For each $k$, $\{D=k\}={\bigcap }_{j<k}\{T_k<T_j\}\cap {\bigcap }_{j>k}\{T_k\le T_j\},$ which is measurable; the assumed almost-sure uniqueness makes the tie-break irrelevant except on a null set. No independence among the latent times $T_j$ is assumed. When an external right-censoring time $C$ is present, the observation is $(U,\delta ,D^{\mathrm{o}\mathrm{b}\mathrm{s}})$, where $U=\min (T,C)$ and $\delta =\mathbf{1}\{T\le C\}$ as in Event-time observation, censoring, and truncation schemes, and $D^{\mathrm{o}\mathrm{b}\mathrm{s}}=D$ when $\delta =1$; the cause label is absent when $\delta =0$. Thus an event tied with censoring is recorded as an event.

The all-cause survival function is $S(t):=\mathbb{P}(T>t)$ for $t\ge 0$. For $u>0$, write $S(u-):=\mathbb{P}(T\ge u)$; this is the left limit of $S$ by continuity from above for measures (Continuity from above when one set has finite measure). Set $S(0-):=1$.

For each cause $k$, its event subdistribution measure and cause-specific cumulative-incidence function are ${\nu }_k(B):=\mathbb{P}(T\in B,D=k),F_k(t):={\nu }_k([0,t])=\mathbb{P}(T\le t,D=k)\mathrm{.}$ for Borel $B\subseteq [0,\mathrm{\infty })$ and $t\ge 0$. The cause-specific hazard measure is the measure with density $w(u):=\{\begin{array}{ll}1\mathrm{/}S(u-),& S(u-)>0,\\ 0,& S(u-)=0\end{array}$ relative to ${\nu }_k$: ${\mathrm{\Lambda }}_k(B):={\int }_{B}w(u){\nu }_k(du),d{\mathrm{\Lambda }}_k(u)=w(u){\nu }_k(du)\mathrm{.}$ The extension of $u\mapsto S(u-)$ by the constant $1$ for $u<0$ is nonincreasing on $\mathbb{R}$ and therefore Borel measurable (Every monotone real function is Borel measurable), so $w$ is a nonnegative Borel function and this measure is defined by The measure with density $f$ relative to $\mu$. It may have infinite total mass; no finite-horizon integrability is assumed in this definition.