Oscillation Phenomenon of Binomial Confidence Intervals

Abstract

One of the most basic and important problems in statistical practice is constructing an interval estimation of the probability of success. The textbook binomial confidence interval that has near universal acceptance is the familiar Wald binomial confidence interval (Wald-z). In recognition that the actual coverage probability of Wald-z is poor for p near 0 or 1, textbooks include a warning that Wald-z should only be used when np 3 5 (or 10). An interesting phenomenon occurs in the actual coverage probability when n is fixed and p I (0, 1). The oscillation of the coverage probability shows that there exist a large number of combinations of n and p that, while satisfying the condition np 3 5, the corresponding coverage probability is considerably smaller than the nominal level This behavior does not disappear even when n is quite large nor when p moves away from the boundaries