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. 

mixture models, pdf, CDF, and intensity functions, GG01, GG91, !!!!!!!!!!

Mixture models frequently are applied to the probability density function, not the cumulative functions. But, based on wikipedia entry on "mixture density" , mixture can be defined in both pdf and CDF, and it seems to interchangeable.  In fact, this is quit natural, because mixture is a linear form. Derivative of linear combinations of CDFs lead to linear combinations of pdfs.

GG01 applied binomial formula to the mortality rate functions, not the viability functions. GG91 used the same approach, page 258-261, section 6.6.

What would happen if we apply binomial formula to the survival functions? Interestingly, GG91 page 265-266, section 6.7, applied binomial formula to the failure CDF function, $F(x)$ in GG01. So, proof changed from GG91 to GG01.

The failure CDF for binomial active change of $q$ of $n$ elements is:
 $ F(x) = (1- q exp(-kx))^n $,
which is a pleasantly clean form. 

In contrast, Witten85 applied weights to the reliability (viability) function, though his formulation of the Gompertz model did not use this approach.

The more that I compare GG91 and GG01, the more it looks like linear rules are the same between viability function and intensity functions (mortality rates). However, I have not been able to prove this myself. In fact, I found it otherwise, somehow. 

Reference:

http://en.wikipedia.org/wiki/Mixture_density