Block, Li, Savits, 2003, initial and final behaviour of failure rate functions for mixtures and systems

The initial approximation of mixture failure rate is  straight forward. For each subpopulation pdf = failure rate *Viability.  Because viability approaches 1 initially, mixture of pdf leads to mixture of failure rates.

Reference:

Initial and Final Behaviour of Failure Rate Functions for Mixtures and Systems
Author(s): Henry W. Block, Yulin Li and Thomas H. Savits Source: Journal of Applied Probability, Vol. 40, No. 3 (Sep., 2003), pp. 721-740 Published by: Applied Probability Trust

Stable URL: http://www.jstor.org/stable/3215946 .

Notes, Lai & Xie, 2006, stochastic ageing and dependence for reliability

page 48, mixture failure rate r(t):
 r(t) = weighted sum of subpopulation pdf / weighted sum of subpopulation survival function
 This can be found by the definition of r(t).  Failure rate is mortality rate in biology.

page 57, initial failure mixture rate r(0+):
 r(0+) = weighted sum of subpopulation r_i(0+),
shown by Block, Li, Savit, 2003, initial and final behavior of failure rate function for mixtures and systems, Journal of Applied Probability, 40, 721-740
 I am surprised that this basic formula has a 2003 citation.  This formula was in GG91 book, and they cited the Barlow, Proschan, (Hunter) book.