  We consider the problem of estimating a continuous distribution function $F$,
as well as meaningful functions $\tau(F)$ under a large class of loss
functions. We obtain best invariant estimators and establish their minimaxity
for H\"{o}lder continuous $\tau$'s and strict bowl-shaped losses with a bounded
derivative. We also introduce and motivate the use of integrated balanced loss
functions which combine the criteria of an integrated distance between a
decision $d$ and $F$, with the proximity of $d$ with a target estimator $d_0$.
Moreover, we show how the risk analysis of procedures under such an integrated
balanced loss relates to a dual risk analysis under an "unbalanced" loss, and
we derive best invariant estimators, minimax estimators, risk comparisons,
dominance and inadmissibility results. Finally, we expand on various
illustrations and applications relative to maxima-nomination sampling,
median-nomination sampling, and a case study related to bilirubin levels in the
blood of babies suffering from jaundice.
