  Standard QCD resummation techniques provide precise predictions for the
spectrum and the cumulant of a given observable. The integrated spectrum and
the cumulant differ by higher-order terms which, however, can be numerically
significant. In this paper we propose a method, which we call the
$\sigma\text{-improved}$ scheme, to resolve this issue. It consists of two
steps: (i) include higher-order terms in the spectrum to improve the agreement
with the cumulant central value, and (ii) employ profile scales that encode
correlations between different points to give robust uncertainty estimates for
the integrated spectrum. We provide a generic algorithm for determining such
profile scales, and show the application to the thrust distribution in $e^+e^-$
collisions at NLL$'$+NLO and NNLL$'$+NNLO.
