= Additive excess-hazard mixture likelihood
Suppose an unobserved class has probabilities $\pi$ and $1-\pi$, with <hazard functions> $q$ and $q+r$, and set $Q(t)=\int_0^tq$, $R(t)=\int_0^tr$. For an independently right-censored observation $(x,v)$, its survival-parameter likelihood contribution is
$$
L=e^{-Q(x)}\{\pi q(x)^v+(1-\pi)[q(x)+r(x)]^ve^{-R(x)}\}.
$$
This 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
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