= Competing end-of-life criteria
<Competing end-of-life criteria> are two or more conditions, any one of which can terminate useful operation under the stated specification. They define a <remaining useful life> by the earliest relevant crossing. If the prospective crossing times, measured from the present, are $T_1,\ldots,T_n$, then
$$
T=\min(T_1,\ldots,T_n).
$$
This is the event structure studied by <competing risks>. A condition ending one operating cycle must not inadvertently be treated as the component's aging endpoint: NASA's battery experiment separately defines a terminal-potential discharge cutoff and a30% rated-capacity-loss aging criterion.<Li-ion Battery Aging Datasets by NASA>[Dataset description, discharge termination versus aging end-of-life threshold]{ref}
For an original mechanical example, suppose a component can reach either an unacceptable vibration level in12h or an unacceptable temperature in20h under a specified operating scenario. If either condition stops operation, its predicted useful life is12h. Requiring both conditions would instead define another stopping rule. An observed early crossing also does not establish when the other criterion would have been reached if the test had continued.
For uncertain crossing times, the <survival function> of useful life is
$$
\Pr(T>t)=\Pr(T_1>t,\ldots,T_n>t).
$$
Only under independence does this equal the product of the individual <survival functions>. For an explicitly hypothetical independent pair of <exponential distributions> with rates $0.01\,\mathrm{h^{-1}}$ and $0.02\,\mathrm{h^{-1}}$, the product is $e^{-0.03t}$ when $t$ is measured in hours. The expected first crossing is $1/0.03\approx33.33\,\mathrm h$, smaller than either individual mean,100h or50h.
The independence assumption is substantive. Shared temperature, load or damage can correlate crossing times, so multiplying marginal <survival functions> can give the wrong answer. Moreover, an intervention triggered by the first crossing can change the subsequent trajectory. The physical <end-of-life criterion>, dependence model and assumed future operation therefore belong with any numerical lifetime prediction.
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