  Given an IID sample from a positive distribution, we provide a method for
constructing rigorous finite sample lower confidence bounds for the expectation
of the distribution. The method is based on constructing rigorous confidence
regions for the cdf of the distribution. We provide some analysis of the
asymptotical behavior of the rigorous LCBs. We apply the method to obtain an
LCB for a particular, controversial, empirical data set, where the validity of
standard methods has been called into question.
