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Abstract:
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We consider confidence intervals for the hypergeometric parameter M, which may be thought of as the number of special items in a population of N items, based on the number X of special items in a sample without replacement. Although this is a classical problem, length-optimal confidence intervals for M were only recently found by W. Wang (JASA, 2015). W. Wang’s intervals are computationally intensive, and to produce an interval for a given X and N, require the calculation of all the intervals for all possible X and M. A less computationally expensive approach is to invert hypothesis tests, which results in size-optimal confidence sets but these are not always intervals. However, we show that the acceptance regions of these tests may always be shifted to make them monotonic while maintaining the significance level, thus producing proper intervals upon inversion. The result is computationally inexpensive, length-optimal confidence intervals for M. We illustrate these intervals on fine particulate matter (PM2.5) data from five Chinese cities. This is joint work with Jay Bartroff (USC) and Gary Lorden (Caltech).
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