Multiple Testing Versus Multiple Estimation. Improper Confidence Sets. Estimation of Directions and Ratios

Abstract

The "$S$-method" of multiple comparison ([5]; [6], Section 3.5) was intended for multiple estimation, possibly combined with multiple testing. It is shown that if only multiple testing is desired a certain "modified $S$-method" is more powerful. While this result is of some theoretical interest, it is recommended after a discussion of the relative advantages of the two methods, that the new one generally not be used in applications. The multiple testing problems considered are related to estimating the direction of a vector or its unoriented direction-estimation problems which also have an inherent interest. A confidence set for a parameter point is called improper if the probability that it gives a trivially true statement is positive. The problems of estimating the direction and unoriented direction of a vector are reformulated to permit solution by proper confidence sets. In the case of the unoriented direction of a $q$-dimensional vector the confidence sets yield solutions of the problem of joint estimation of $q - 1$ ratios and the problem of multiple estimation of all ratios in a certain infinite set. Specializing to the case $q = 2$ yields a proper confidence set as a substitute for Fieller's improper confidence set for a ratio.