On Simultaneous Confidence Intervals for Multinomial Proportions

In this article we present a method for obtaining simultaneous confidence intervals for the parameters of a multinomial distribution, and we compare this method with the one suggested recently by Quesenberry and Hurst (1964). For the usual probability levels, we find, for example, that the confidence intervals introduced here have the desirable property that they are shorter than the corresponding intervals obtained by the Quesenberry-Hurst method. We also present methods for obtaining simultaneous confidence intervals for the differences among the parameters of the multinomial distribution, and we compare these methods with the one suggested earlier by Gold (1963) for studying linear functions of the multinomial parameters. For the usual probability levels, we find that the confidence intervals introduced in the present article have the desirable property that they are shorter than the corresponding intervals obtained by the Gold method applied to the differences among the multinomial parameters. In addition, we show how the methods presented here for studying the differences among the multinomial parameters can be modified in order to obtain simultaneous confidence intervals for the relative differences among the multinomial parameters.