A simple algorithm for tighter exact upper confidence bounds with rare attributes in finite universes

When attributes are rare and few or none are observed in the selected sample from a finite universe, sampling statisticians are increasingly being challenged to use whatever methods are available to declare with high probability or confidence that the universe is near or completely attribute-free. This is especially true when the attribute is undesirable. Approximations such as those based on normal theory are frequently inadequate with rare attributes. For simple random sampling without replacement, an appropriate probability distribution for statistical inference is the hypergeometric distribution. But even with the hypergeometric distribution, the investigator is limited from making claims of attribute-free with high confidence unless the sample size is quite large using nonrandomized techniques. For students in statistical theory, this short article seeks to revive the question of the relevance of randomized methods. When comparing methods for construction of confidence bounds in discrete settings, randomization methods are useful in fixing like confidence levels and hence facilitating the comparisons. Under simple random sampling, this article defines and presents a simple algorithm for the construction of exact "randomized" upper confidence bounds which permit one to possibly report tighter bounds than those exact bounds obtained using "nonrandomized" methods. A general theory for exact randomized confidence bounds is presented in Lehmann (1959, p. 81), but Lehmann's development requires more mathematical development than is required in this application. Not only is the development of these "randomized" bounds in this paper elementary, but their desirable properties and their link with the usual nonrandomized bounds are easy to see with the presented approach which leads to the same results as would be obtained using the method of Lehmann.