  In this paper, a version of the Polyakov-Nambu-Jona-Lasinio (PNJL) model
based on nonextensive statistical mechanics is presented. This new statistics
summarizes all possible factors that violate the assumptions of the
Boltzmann-Gibbs (BG) statistics to a dimensionless nonextensivity parameter
$q$, and when $q$ tends to 1, it returns to the BG case. Within the
nonextensive PNJL model, we found that as $q$ increases, the location of the
critical end point (CEP) exhibits non-monotonic behavior. That is, for
$q<1.15$, CEP moves in the direction of lower temperature and larger quark
chemical potential. But for $q>1.15$, CEP turns to move in the direction of
lower temperature and lower quark chemical potential. In addition, we studied
the moments of the net-baryon number distribution, that is, the variance
($\sigma^{2}$), skewness (S), and kurtosis ($\kappa$). Our results are
generally consistent with the latest experimental data, especially for
$\sqrt{S_{NN}}>19.6\ \mathrm{GeV}$, when $q$ is set to $1.07$.
