Population hazard of a survival mixture ($\overline{h}(t)=\frac{{\sum }_i{F}_i(t)h_i(t)}{{\sum }_i{F}_i(t)}$)

For an equally weighted mixture of individual survivor functions $F_i$, the population hazard function is $\overline{h}(t)={\sum }_i{F}_i(t)h_i(t)/{\sum }_i{F}_i(t)$. Thus the weights are the surviving proportions, and the cumulative hazard function is $-\log (n^{-1}{\sum }_i{e}^{-H_i(t)})$, not generally the mean individual integrated hazard.

In a population mixture with initial component probabilities $p_i$, conditioning on survival to time $t$ changes them to the displayed probabilities by Bayes theorem. Components with larger survivor functions receive greater relative weight among survivors. The population hazard function is the weighted mean of the component hazards with these time-dependent weights. This selection can change the population hazard even when all component hazards are constant, as shown by the hazard derivative for a mixture of exponential distributions.

For a finite mixture with positive weights $p_i$ and rates ${\lambda }_i>0$, the survivor function is $S(t)={\sum }_i{p}_i{e}^{-{\lambda }_{i}t}$ and the hazard function is the surviving-population mean of the component rates. Write $w_i(t)=p_i{e}^{-{\lambda }_{i}t}/S(t)$. Differentiation gives $w_i^{\prime }=w_i(h-{\lambda }_i)$ and therefore $h^{\prime }=h^2-{\sum }_i{w}_i{\lambda }_i^2=-\mathrm{Var}(\Lambda \mid T>t)$. The hazard is decreasing, strictly so when distinct rates have positive weights. For finitely many rates it tends to the smallest rate with positive weight. This is survival selection even though each individual component has a constant hazard.