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

A rigorous lower confidence bound for the expectation of a positive random variable