  It has been recently empirically established that some of the directly
observed pa- rameters of GRBs are correlated with their important intrinsic
parameters, like the luminosity or the total radiated energy. These
correlations were derived, tested and used to standardize GRBs, i.e., to derive
their luminosity or radiated energy from one or more observables, in order to
construct an estimated fiducial Hubble diagram, assuming that radiation
propagates in the standard LambdaCDM cosmological model. We extend these
analyses by considering more general models of dark energy, and an updated data
set of high redshift GRBs. We show that the correlation parameters only weakly
depend on the cosmological model. Moreover we apply a local regression
technique to estimate, in a model independent way, the distance modulus from
the recently updated SNIa sample containing 307 SNIa (Astier et al. 2006), in
order to calibrate the GRBs 2D correlations, considering only GRBs with z <1.4.
The derived calibration parameters are used to construct a new GRBs Hubble
diagram, which we call the calibrated GRBs HD. We also compare the estimated
and calibrated GRBs HDs. It turns out that for the common GRBs they are fully
statistically consistent, thus indicating that both of them are not affected by
any systematic bias induced by the different standardizing procedures. We
finally apply our methods to calibrate 95 long GRBs with the well-known Amati
relation and construct the estimated and calibrated GRBs Hubble diagram that
extends to redshifts z ~ 8. Even in this case there is consistency between
these datasets. This means that the high redshift GRBs can be used to test
different models of dark energy. We used the calibrated GRBs HD to constrain
our quintessential cosmological model and derived the likelihood values of
Omega_m and w(0).
