  Although today there are many observational methods, Type Ia supernovae
(SNIa) is still one of the most powerful tools to probe the mysterious dark
energy (DE). The most recent SNIa datasets are the 307 SNIa "Union" dataset
\cite{kow08} and the 397 SNIa "Constitution" dataset \cite{hic09}. In a recent
work \cite{wei10}, Wei pointed out that both Union and Constitution datasets
are in tension with the observations of cosmic microwave background (CMB) and
baryon acoustic oscillation (BAO), and suggested that two truncated versions of
Union and Constitution datasets, namely "UnionT" and "ConstitutionT", should be
used to constrain various DE models. But in \cite{wei10}, only the $\Lambda$CDM
model is used to select the outliers from the Union and the Constitution
dataset. In principle, since different DE models may select different outliers,
the truncation procedure should be performed for each different DE model. In
the present work, by performing the truncation procedure of \cite{wei10} for 10
different models, we demonstrate that the impact of different models is
negligible, and the approach adopted in \cite{wei10} is valid. Moreover, by
using the 4 SNIa datasets mentioned above, as well as the observations of CMB
and BAO, we perform best-fit analysis on the 10 models. It is found that: (1)
For each DE model, the truncated SNIa datasets not only greatly reduce
$\chi_{min}^{2}$ and $\chi_{min}^{2}/dof$, but also remove the tension between
SNIa data and other cosmological observations. (2) The CMB data is very helpful
to break the degeneracy among different parameters, and plays a very important
role in distinguishing different DE models. (3) The current observational data
are still too limited to distinguish all DE models.
