  We compare estimators of the (essential) supremum and the integral of a
function $f$ defined on a measurable space when $f$ may be observed at a sample
of points in its domain, possibly with error. The estimators compared vary in
their levels of stratification of the domain, with the result that more refined
stratification is better with respect to different criteria. The emphasis is on
criteria related to stochastic orders. For example, rather than compare
estimators of the integral of $f$ by their variances (for unbiased estimators),
or mean square error, we attempt the stronger comparison of convex order when
possible. For the supremum, the criterion is based on the stochastic order of
estimators.
