Right-censored exponential likelihood and its boundary maximum

For any Borel $B\subseteq [0,\mathrm{\infty })$, the event branch satisfies $P(U\in B,\delta =1)=\int \text{\hspace{0.17em}⁣}\int 1\{t\in B,t\le c\}\lambda e^{-\lambda t}dt{G}_{\psi }(dc)={\int }_B\lambda e^{-\lambda u}{G}_{\psi }([u,\mathrm{\infty }))du\mathrm{.}$ Here we use the event-density version $\lambda e^{-\lambda t}$ for $t\ge 0$, which differs from [F2] only at the Lebesgue-null singleton $\{0\}$ by [F11]. The indicator is product measurable because $B$ is Borel and $\{(t,c):t\le c\}$ is closed. The factor $\lambda e^{-\lambda t}$ is continuous, so [F17] makes the full integrand product measurable. The first equality uses the product law from [A3], [F2], and [F3]; the second integrates the nonnegative indicator in the censor coordinate by Tonelli.

On the censor branch, $U=C$ and $T>C$, so $P(U\in B,\delta =0)={\int }_{B}P(T>u)G_{\psi }(du)={\int }_B{e}^{-\lambda u}{g}_{\psi }(u)\nu (du)\mathrm{.}$ The integrand $1_B(u)e^{-\lambda u}$ is measurable by [F17]. This uses the survival in [F2], independence, and the specified density of $G_{\psi }$; [F12] changes integration against $G_{\psi }=g_{\psi }\nu$ into integration against $g_{\psi }d\nu$.

The formal survival boundary $S_0(t)=1$ cannot be a probability law of a finite-valued event time: for every integer $m$, it would give $P(T\le m)=0$, while $[0,\mathrm{\infty })={\bigcup }_m[0,m]$ and [F7] would then give $P(T\in [0,\mathrm{\infty }))=0$. Thus the boundary value is a likelihood extension unless the event-time model is enlarged to admit $T=\mathrm{\infty }$.

Extend $u\mapsto {\bar{G}}_{\psi }(u)$ to $u<0$ by the constant $G_{\psi }([0,\mathrm{\infty }))=1$. This extension is nonincreasing by inclusion of the tail sets, so [F9] makes it Borel measurable. The event-branch density is measurable by [F17], and the censor-branch density is measurable by [F17] because $g_{\psi }$ is measurable. The event branch is sigma-finite because [F11] gives the finite exhaustion $[0,n]$; the censor branch uses a countable finite-measure exhaustion of $\nu$. Their union across the disjoint branches makes $\rho$ sigma-finite. Steps 1.1 and 1.2 then show that the one-subject observation law is dominated by $\rho$ with the displayed density $h_{\lambda ,\psi }$.

For each $i$, $(T_i,C_i)$ is a measurable map into the product space by [F3]. The map $(t,c)\mapsto (\min (t,c),1\{t\le c\})$ is Borel: minimum is continuous and $\{(t,c):t\le c\}$ is closed. Thus [F10] makes each observed pair measurable under the convention [F1]; taking the finite tuple is measurable since rectangle preimages are finite intersections. The coordinate pullbacks of $h_{\lambda ,\psi }$ are measurable, so repeated use of [F17] makes $D(\mathbf{u},\mathit{\delta }):={\prod }_{i=1}^n{h}_{\lambda ,\psi }(u_i,{\delta }_i)$ measurable. Let $Q$ be its density measure relative to ${\rho }^{\otimes n}$, which exists by [F16]. By step 2.1, each one-subject density integrates to one. Repeated Tonelli integration therefore makes $Q$ a probability measure and gives, for every measurable rectangle $R={\prod }_{i=1}^n{B}_i$, $Q(R)={\prod }_{i=1}^n{\int }_{B_i}{h}_{\lambda ,\psi }d\rho \mathrm{.}$ Independence of the subject pairs and [F3] give the same rectangle probabilities for their joint observation law. Measurable rectangles form a generating $\pi$-system, and the whole product space is a finite-measure exhaustion for both probability measures. The uniqueness clause of [F3], iterated over the $n$ factors, now identifies $Q$ with the joint observation law. Hence $D$ is the asserted density version.

Substituting the two branch formulas from step 2.1 into the product density of step 3.1 gives $L(\lambda ,\psi ;\mathbf{u},\mathit{\delta })={\lambda }^{{\sum }_i{\delta }_i}{e}^{-\lambda {\sum }_i{u}_i}{\prod }_{i:{\delta }_i=1}{\bar{G}}_{\psi }(u_i){\prod }_{i:{\delta }_i=0}{g}_{\psi }(u_i)\mathrm{.}$ By [F5], the experiment and parameter map give the sampling-law context; under the parameter-distinct model [A2], this is the asserted factorization. The density representatives are explicit model inputs, so no arbitrary simultaneous selection of Radon–Nikodym versions is used; the equality is stated separately almost everywhere for each fixed parameter pair.

Fix $\psi$ with $C_{\psi }>0$. For $d>0$ and $E>0$, the log likelihood up to an additive constant is $\mathrm{\ell }(\lambda )=d\log \lambda -E\lambda$. Its derivative is $d\mathrm{/}\lambda -E$, which is positive for $0<\lambda <d\mathrm{/}E$ and negative for $\lambda >d\mathrm{/}E$. By [F13] it is strictly increasing before $d\mathrm{/}E$ and strictly decreasing after it, so its unique global maximum is at $\widehat{\lambda }=d\mathrm{/}E$.

If $d=0$ and $E>0$, then the event factor is $e^{-\lambda E}$, strictly decreasing on $(0,\mathrm{\infty })$ by [F14], and tends to $1$ as $\lambda \downarrow 0$ by [F8]. The supremum is therefore not attained in the ordinary parameter space. If the likelihood alone is extended to include $\lambda =0$, it is attained there.

If $d=0$ and $E=0$, the event factor is identically $1$, so every $\lambda >0$ maximizes it. If $d>0$ and $E=0$, some event has $U_i=T_i=0$; but the exponential law has $P(T_i=0)=0$, so such observations form a probability-zero boundary set.

Steps 1.1, 1.2, 2.1, 3.1, and 4.1 prove the dominated likelihood factorization, and steps 5.1–5.3 establish all stated interior, boundary, and zero-exposure cases.