Large Sample Simultaneous Confidence Intervals for the Multinomial Probabilities Based on Transformations of the Cell Frequencies

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Abstract
The paper outlines a detailed study of three sets of large sample simultaneous confidence intervals for the probabilities of a multinomial distribution, all three making use of the Bonferroni inequality. One of them, originally proposed by Goodman, is based on the assumption that each cell frequency ni is, marginally, normally distributed, while the other two require the normality of transformations of ni —an angular transformation in one case and a square root in the other. It is shown that all three sets of intervals should be used with a correction for continuity; their coverage probabilities are investigated, and it is seen that the two sets based on transformations of ni produce shorter intervals than Goodman's when ni is small. There is little to choose between these two except that one of them is a little simpler to use than the other.

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