= Hazard derivative for a mixture of exponential distributions
{title2=$h'(t)=-\operatorname{Var}(\Lambda\mid T>t)$}
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 $w_i'=w_i(h-\lambda_i)$ and therefore $h'=h^2-\sum_iw_i\lambda_i^2=-\operatorname{Var}(\Lambda\mid T>t)$. 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.
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