  We study the problem of finding the index of the minimum value of a vector
from noisy observations. This problem is relevant in population/policy
comparison, discrete maximum likelihood, and model selection. We develop a test
statistic that is asymptotically normal, even in high-dimensional settings and
with potentially many ties in the population mean vector, by integrating
concepts and tools from cross-validation and differential privacy. The key
technical ingredient is a central limit theorem for globally dependent data. We
also propose practical ways to select the tuning parameter that adapts to the
signal landscape.
