  The non-Abelian topological order has attracted a lot of attention for its
fundamental importance and exciting prospect of topological quantum
computation. However, explicit demonstration or identification of the
non-Abelian states and the associated statistics in a microscopic model is very
challenging. Here, based on density-matrix renormalization group calculation,
we provide a complete characterization of the universal properties of bosonic
Moore-Read state on Haldane honeycomb lattice model at filling number $\nu=1$
for larger systems, including both the edge spectrum and the bulk anyonic
quasiparticle (QP) statistics. We first demonstrate that there are three
degenerating ground states, for each of which there is a definite anyonic flux
threading through the cylinder. We identify the nontrivial countings for the
entanglement spectrum in accordance with the corresponding conformal field
theory. Through inserting the $U(1)$ charge flux, it is found that two of the
ground states can be adiabatically connected through a fermionic
charge-$\textit{e}$ QP being pumped from one edge to the other, while the
ground state in Ising anyon sector evolves back to itself. Furthermore, we
calculate the modular matrices $\mathcal{S}$ and $\mathcal{U}$, which contain
all the information for the anyonic QPs. In particular, the extracted quantum
dimensions, fusion rule and topological spins from modular matrices positively
identify the emergence of non-Abelian statistics following the $SU(2)_2$
Chern-Simons theory.
