  Ranks estimated from data are uncertain and this poses a challenge in many
applications. However, estimated ranks are deterministic functions of estimated
parameters, so the uncertainty in the ranks must be determined by the
uncertainty in the parameter estimates. We give a complete characterization of
this relationship in terms of the linear extensions of a partial order
determined by interval estimates of the parameters of interest. We then use
this relationship to give a set estimator for the overall ranking, use its size
to measure the uncertainty in a ranking, and give efficient algorithms for
several questions of interest. We show that our set estimator is a valid
confidence set and describe its relationship to a joint confidence set for
ranks recently proposed by Klein, Wright \& Wieczorek. We apply our methods to
both simulated and real data and make them available through the R package
rankUncertainty.
