 Additive excess-hazard mixture likelihood

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