Event-time observation, censoring, and truncation schemes

An event time is a measurable real-valued map $T$ with values in $[0,\mathrm{\infty })$. A covariate is a measurable map $Z$ into a specified standard-Borel space $(\mathcal{Z},\mathcal{B})$, where standard Borel means measurably isomorphic to a Polish space (Standard Borel spaces). A censoring or inspection time is a measurable real-valued map $C$ with values in $[0,\mathrm{\infty })$. When conditional laws are used, they are probability kernels $K_T(z,\cdot )$ and $K_C(z,\cdot )$ in the sense of Conditional law given a random element and Measure kernel and probability kernel. They are part of the specified model; their existence for real-valued targets and factorization through standard-Borel $Z$ follow under The Axiom of Choice from Existence of regular conditional distributions for standard borel targets and Regular conditional kernels factor through a standard borel conditioning variable.

The notation $T\perp \text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\perp C\mid \sigma (Z)$ has the meaning in Conditional independence given a sigma-algebra: for bounded measurable $a,b$, $\mathbb{E}[a(T)b(C)\mid \sigma (Z)]=\mathbb{E}[a(T)\mid \sigma (Z)]\mathbb{E}[b(C)\mid \sigma (Z)]$ almost surely. It is a model assumption about the conditional observation law, not a consequence of adjusting for $Z$ and not an empirical conclusion.

Under right censoring, the recorded pair is $U=\min (T,C)$ and $\delta =1\{T\le C\}$. Thus $\delta =1$ records an event at $U=T$, including a tie, while $\delta =0$ records censoring at $U=C$ and implies $T>C$. Left censoring at a known threshold $L$ records only $T\le L$. Interval censoring at known endpoints $L<R$ records $L<T\le R$. Current-status observation at an inspection time $C$ records only ${\delta }_C=1\{T\le C\}$.

Left truncation, also called delayed entry at $L$, selects a unit into the sample only when $T>L$; observation begins after that selection. Right truncation at $R$ selects a unit only when $T<R$. These are selection rules before the observed sample is formed, whereas censoring coarsens an event time after sampling. The strict and weak inequalities in these definitions fix the endpoint conventions. A random observation endpoint is called noninformative here only when it is fixed by design or is generated independently of $T$ conditional on the stated covariates under a parameter-distinct law; otherwise its joint observation law must be modeled.