Philip Stark
UC Berkeley
Simultaneous Confidence Intervals with more Power to Determine Signs
P.B. Stark. Joint work with Y. Benjamini and V. Madar
Equivariance, optimality, unbiasedness, multiplicity, and the duality
between tests and confidence sets are among the topics Erich Lehmann
studied and addressed so beautifully in his books "Theory of Point
Estimation" and "Testing Statistical Hypotheses." I will present a new
method that exploits the duality between tests and confidence
sets--but sacrifices equivariance and unbiasedness--to construct
simultaneous confidence intervals for the components of a multivariate
mean that determine the signs of the parameters more frequently than
standard translation-equivariant intervals do. When one or more
estimated means are small, the new intervals sacrifice some length to
avoid crossing zero. But when all the estimated means are large, the
new intervals coincide with standard, equivariant, simultaneous
confidence intervals, so there is no loss of precision. The
improvement can be substantial. For example, if two means are to be
estimated and the intervals are allowed to be at most 80% longer than
standard intervals in the worst case, then when only one mean is small
its sign is determined almost as well as by a one-sided test that
ignores multiplicity and has a pre-specified direction. When both are
small the sign is determined better than by two-sided tests that
ignore multiplicity. The intervals are constructed by inverting
level-$\alpha$ tests to form a $1-\alpha$ confidence set, then
projecting that set onto the coordinate axes to get confidence
intervals. The tests have hyperrectangular acceptance regions that
minimize the maximum amount by which the acceptance region protrudes
from the orthant that contains the hypothesized parameter value,
subject to a constraint on the maximum side length of the
hyperrectangle. These tests are biased in general and the resulting
confidence sets are equivariant under permutations of the coordinates
and sign changes, but not under translation.