Related Inequalities

As noted by Pólya (1967), “Inequalities play a role in most branches of mathematics and have widely different applications.” This is certainly true in statistics and probability. From the viewpoint of applications, inequalities have become a useful tool in estimation and hypothesis-testing problems (such as for yielding bounds on the variances of estimators and on the probability contents of confidence regions, and for establishing monotonicity properties of the power functions of certain tests), in multivariate analysis, in reliability theory, and so forth. Perhaps the usefulness of inequalities in multivariate analysis can be best illustrated by the following situation: Suppose that in an applied problem the confidence probability of a given confidence region for the mean vector is difficult to evaluate. If an inequality in the form of a lower bound on the confidence probability can easily be obtained, and if the lower bound already meets the required level of specification, then we know for sure that the true confidence probability meets or exceeds the required level.