Likelihood contributions under censoring, truncation, and delayed entry

For fixed $(\theta ,\eta )$, $[0,\mathrm{\infty })$ is closed in $\mathbb{R}$, and the max metric makes $[0,\mathrm{\infty }{)}^2$ complete: a Cauchy sequence is Cauchy in each real coordinate, converges by completeness of $\mathbb{R}$, and its coordinate limits remain nonnegative. For $0\le x<y$, the Borel ray $(x,\mathrm{\infty })$ separates $x$ and $y$, so the rational-ray generators in [F7] include some $(q,\mathrm{\infty })$ that separates them, giving $x\le q<y$. Taking $y=x+\epsilon$ gives rational approximations from above; applying this to both coordinates shows the countable set $(\mathbb{Q}\cap [0,\mathrm{\infty }){)}^2$ is dense, with countability from [F7]. Thus $[0,\mathrm{\infty })$ and its square are Polish and standard Borel. By [F5] and [A1], take conditional-law kernels for $T$, $C$, and $(T,C)$ given $Z$. For each pair of rational half-lines $A,B$, [F6] applied to their indicators and the conditional-law identities in [F4] give $K_{T,C}(Z,A\times B)=K_T(Z,A)K_C(Z,B)$ almost surely.

If $L=R$, then $(L,R]$ is empty and both its probability and $F_{\theta }(R\mid z)-F_{\theta }(L\mid z)$ are zero. If $L<R$, the conditional event law is a Borel probability, hence finite on compact sets, and [F9] gives ${\mu }_{\theta ,z}((L,R])=F_{\theta }(R\mid z)-F_{\theta }(L\mid z)$. The singleton formula in [F9] gives ${\mu }_{\theta ,z}(\{t\})=F_{\theta }(t\mid z)-F_{\theta }(t^{-}\mid z)$.

By the definition of $F_{\theta }$, $P(T\le L\mid z)=F_{\theta }(L\mid z)$ and $P(T>c\mid z)=S_{\theta }(c\mid z)=1-F_{\theta }(c\mid z)$. Splitting the two values of $\delta =1\{T\le c\}$ gives the current-status contribution. For random inspection endpoints conditionally independent of $T$ given $Z$, conditioning on the endpoints leaves these event probabilities unchanged, and their law has no $\theta$ factor.

Since $T\ge 0$ and the selected probability is positive, $R>0$. The open-right interval formula in [F9] gives $P(T<R\mid z)=F_{\theta }(R^{-}\mid z)$. Conditioning the event law on $T<R$ divides its restriction to $(-\mathrm{\infty },R)$ by this positive probability; for an observed threshold independent of $T$ given $Z$, the same conditional law applies given the threshold. When the original law has density, the selected density is $f_{\theta }(t\mid z)\mathrm{/}{F}_{\theta }(R^{-}\mid z)$ for $t<R$. This uses a selection normalization and has no right-censor factor.

For $u\ge 0$, put $a_n(u)=2^{-n}\lceil 2^{n}u\rceil$ and $b_n(u)=2^{-n}\lfloor 2^{n}u\rfloor$. For each $n$, the maps $(z,u)\mapsto K_{\theta }^T(z,(a_n(u),\mathrm{\infty }))$ and $(z,u)\mapsto K_{\eta }^C(z,[b_n(u),\mathrm{\infty }))$ are measurable: the threshold maps are countably valued Borel functions and each fixed rational-ray evaluation is measurable in $z$ by [F4]. As $a_n(u)\downarrow u$, the first ray sets increase to $(u,\mathrm{\infty })$; as $b_n(u)\uparrow u$, the second ray sets decrease to $[u,\mathrm{\infty })$. Countable additivity gives continuity from below and, for these probability sections, from above. Thus the maps converge pointwise to $S_{\theta }(u\mid z)$ and $G_{\eta }(u\mid z)$, so both tails are jointly measurable.

The rational rectangles form a countable pi-system generating the product Borel sigma-algebra by [F7]. Removing the countable union of their exceptional null sets makes the equality in step 1.1 simultaneous on this pi-system. The product measure exists by [F8], so [F7] gives $K_{T,C}(z,\cdot )=K_T(z,\cdot )\otimes K_C(z,\cdot )$ for almost every $z$.

Under this product law, for every Borel $B\subseteq [0,\mathrm{\infty })$, Tonelli and the density of $K_{\theta }^T$ give $P(U\in B,\delta =1\mid z)={\int }_B{K}_{\eta }^C(z,[t,\mathrm{\infty }))f_{\theta }(t\mid z)dt$. Thus the density on the $\delta =1$ branch with respect to Lebesgue measure is $f_{\theta }(u\mid z)G_{\eta }(u\mid z)$; it is jointly measurable in $(z,u)$ by step 1.5 and the assumed measurability of $f_{\theta }$.

On the $\delta =0$ branch, $T>C$ and $U=C$. Tonelli gives $P(U\in B,\delta =0\mid z)={\int }_B{K}_{\theta }^T(z,(c,\mathrm{\infty }))K_{\eta }^C(z,dc)={\int }_B{S}_{\theta }(c\mid z)g_{\eta }(z,c)\nu (dc)$, and the resulting density is jointly measurable by step 1.5 and the assumed measurability of $g_{\eta }$.

The two branch measures are dominated by $\rho$, so the displayed functions are their densities. The recorded object is one subject's $(Z,U,\delta )$ and the likelihood is conditional on $Z$, as explicitly declared in [F10]; this bookkeeping adds no scientific truth claim. The parametric experiment and likelihood conventions [F2]–[F3] therefore yield the stated likelihood. Because the parameter space is $\mathrm{\Theta }\times H$ and the event and censor laws depend on $\theta$ and $\eta$ separately, the factors $f_{\theta }^{\delta }{S}_{\theta }^{1-\delta }$ and $G_{\eta }^{\delta }{g}_{\eta }^{1-\delta }$ separate; only the former depends on $\theta$.

For fixed entry, the event kernel given the pre-entry covariates $W$ is $K_{\theta }^T(W,\cdot )$. For random entry, the stated factorization of the pre-entry conditional joint law gives the same event kernel given $Z=(W,L)$ before selection. In either case, conditional on $Z$ and the positive-probability inclusion event $T>L$, the event law assigns a set $A$ probability $K_{\theta }^T(W,A\cap (L,\mathrm{\infty }))\mathrm{/}{S}_{\theta }(L\mid W)$. Conditional independence of $T$ and $C$ given $Z$ means restricting their product law to $T>L$ leaves this truncated event law and the same censor law as a product. The event density is therefore $f_{\theta }(u\mid W)\mathrm{/}{S}_{\theta }(L\mid W)$ for $u>L$, and the conditional survivor is $S_{\theta }(u\mid W)\mathrm{/}{S}_{\theta }(L\mid W)$ for every $u\ge L$; at $u=L$ this equals $1$, including when the censoring law has an atom at entry. Applying the branch calculations from steps 3.1–4.1 under that product law yields the asserted denominator, once, in either event-status branch.

Every factorization above was conditional on the stated independence, domination, endpoint, and positivity assumptions. The calculations give no evidence that a real censoring, inspection, or entry mechanism satisfies them; an informative mechanism requires its own joint law.