Estimation of the Means of Dependent Variables

Abstract

Methods are given for constructing sets of simultaneous confidence intervals for the means of variables which follow a multivariate normal distribution. In section (3), a set of confidence intervals is obtained for each of two special cases; first when the variances are assumed to be known, and second when the variances are assumed to be equal. These two sets have the property that the confidence is known exactly, rather than merely being bounded below. In the case of known variances, the intervals are of fixed lengths (i.e., the lengths are the same from sample to sample); when the variances are unknown, the intervals are of variable lengths. It may be surprising to note that nothing need be known about the covariances in order to obtain confidence intervals of fixed lengths whose confidence coefficient is exact. These intervals are long, and do not make use of all the information provided by the sample. Each of sections (4) to (7) considers a different method for obtaining confidence intervals of bounded confidence level. In each section a set of fixed lengths is obtained when the variances are assumed to be known, while a set of variable lengths is obtained when the variances are unknown but equal. In section (5) the set of variable lengths applies to the general multivariate normal distribution, all the other confidence intervals in this paper require some assumption concerning the variances. In section (8) the sets of intervals are compared on the basis of length. One of the bounded confidence level methods, which has been established only for two or three variables or for an arbitrary number of variables with a special type of correlation matrix, is shown to yield the best possible set. Another of the bounded confidence level methods, whose use is established in general, is shown to be almost as good as the best set for confidence coefficients of practical interest. It is interesting to notice that intervals with bounded confidence level, are found which are much shorter than the ones whose confidence level is exact. This need not surprise us, however. In the case of just one variable, we might easily find hat the 95% confidence intervals for the mean using the $t$-statistic were shorter on the average than 94% confidence intervals using order statistics. Moreover, since in admitting sets of confidence intervals with bounded confidence level we consider a much broader class of methods, we might almost expect that some of them would give better intervals.