  We consider the further development of the formalism of the estimates of
higher-order perturbative corrections in the Euclidean region, which is based
on the application of the scheme-invariant methods, namely the principle of
minimal sensitivity and the effective charges approach. We present the
estimates of the order $O(\alpha^{4}_{s})$ QCD corrections to the Euclidean
quantities: the $e^+e^-$-annihilation $D$-function and the deep inelastic
scattering sum rules, namely the non-polarized and polarized Bjorken sum rules
and to the Gross--Llewellyn Smith sum rule. The results for the $D$-function
are further applied to estimate the $O(\alpha_s^4)$ QCD corrections to the
Minkowskian quantities $R(s) = \sigma_{tot} (e^{+}e^{-} \to {\rm hadrons}) /
\sigma (e^{+}e^{-} \to \mu^{+} \mu^{-})$ and $R_{\tau} = \Gamma (\tau \to
\nu_{\tau} + {\rm hadrons}) / \Gamma (\tau \to \nu_{\tau} \overline{\nu}_{e}
e)$. The problem of the fixation of the uncertainties due to the
$O(\alpha_s^5)$ corrections to the considered quantities is also discussed.
