Survival likelihood

With independent right censoring, an exact event at $t$ contributes $h(t)S(t)$ to a survival likelihood, while right censoring at $t$ contributes $S(t)$.

Suppose an unobserved class has probabilities $\pi$ and $1-\pi$, with hazard functions $q$ and $q+r$, and set $Q(t)={\int }_0^{t}q$, $R(t)={\int }_0^{t}r$. For an independently right-censored observation $(x,v)$, its survival-parameter likelihood contribution isThis sums the two class-specific likelihoods before taking a logarithm. All hazards must be nonnegative. A common parameter-free censoring mechanism factors out; class-specific censoring instead alters the mixture weights and cannot silently be discarded.

// Target: probability-and-statistics.bigb

Under independent censoring and a constant event rate, $d$ observed events over total observed person-time $X$ give likelihood proportional to ${\lambda }^d{e}^{-\lambda X}$ and estimate $d/X$ for $d,X>0$. Censored observations belong in $X$. With no events the positive-rate likelihood has a boundary supremum.

Under independent censoring, a model $H_i(t)=\lambda t+G_i(t)$ with known $G_i$ has score ${\sum }_i{v}_i/(\lambda +G_i^{\prime }(x_i))-X$, where $X={\sum }_i{x}_i$ and $d={\sum }_i{v}_i$. Expanding about the ordinary exponential estimate $d/X$ gives the displayed first-order correction when every added event-time hazard is small compared with $d/X$. The integrated offsets contribute constants to the log-likelihood. If all $G_i^{\prime }(x_i)=g$, the positive interior estimate is exactly $d/X-g$.

A likelihood contribution is the probability mass or density assigned by a statistical model to one observation's recorded information.