A method for constructing upper confidence limits (UCL's) for parametric functions of typeg(λ) = ΣjΠi∈Ijλiis considered. It is based on maximization ofg(λ) over a confidence set of type {λ | ΣλiTi≤ Λα& λi≥ 0} obtained by pooling statistical data received from separate testing of system components. An asymptotic representation of the reliability function of an arbitrary coherent system is given assuming that the element failure rates are low. It has the formR(t) ≈ 1 - exp (–trg(λ)) and, therefore, a UCLg* forg(λ) provides a lower confidence limit (LCL) forR(t). Explicit formulas are presented for parallel, series-parallel, andk-out-of-nsystems, for two types of statistical test data, namely, testing over a given time period and testing until a prescribed number of failures appear. The performance of the method is discussed in terms of two characteristics: the average value ofg*/g(λ) and the true confidence coefficientP{g* ≥g(λ)}.