  In this paper, we obtain two improved Amati correlations of the Gamma-Ray
burst (GRB) data via a powerful statistical tool called copula. After
calibrating, with the low-redshift GRB data, the improved Amati correlations
based on a fiducial $\Lambda$CDM model with $\Omega_\mathrm{m0}=0.3$ and
$H_0=70~\mathrm{km~s^{-1}Mpc^{-1}}$, and extrapolating the results to the
high-redshift GRB data, we obtain the Hubble diagram of GRB data points.
Applying these GRB data to constrain the $\Lambda$CDM model, we find that the
improved Amati correlation from copula can give a result well consistent with
$\Omega_\mathrm{m0}=0.3$, while the standard Amati and extended Amati
correlations do not. This results suggest that when the improved Amati
correlation from copula is used in the low-redshift calibration method, the GRB
data can be regarded as a viable cosmological explorer. However, the Bayesian
information criterion indicates that the standard Amati correlation remains to
be favored mildly since it has the least model parameters. Furthermore, once
the simultaneous fitting method rather than the low-redshift calibration one is
used, there is no apparent evidence that the improved Amati correlation is
better than the standard one. Thus, more works need to be done in the future in
order to compare different Amati correlations.

Cosmology and Nongalactic Astrophysics;General Relativity and Quantum Cosmology