  In light of the recent neutrino experiment results from Daya Bay and RENO
Collaborations, we study phenomenology of neutrino mixing angles in the Type
III seesaw model with an discrete $A_4 \times Z_2$ symmetry, whose
spontaneously breaking scale is much higher than the electroweak scale. At tree
level, the tri-bimaximal (TBM) form of the lepton mixing matrix can be obtained
from leptonic Yukawa interactions in a natural way. We introduce all possible
effective dimension-5 operators, invariant under the Standard Model gauge group
and $A_4 \times Z_2$, and explicitly show that they induce a deviation of the
lepton mixing from the TBM mixing matrix, which can explain a large mixing
angle $\theta_{13}$ together with small deviations of the solar and atmospheric
mixing angles from the TBM. Two possible scenarios are investigated, by taking
into account either negligible or sizable contributions from the light charged
lepton sector to the lepton mixing matrix. Especially it is found in the latter
scenario that all the neutrino experimental data, including the recent best-fit
value of $\theta_{13} = 8.68^{\circ}$, can be accommodated. The leptonic CP
violation characterized by the Jarlskog invariant $J_{CP}$ has a non-vanishing
value, indicating a signal of maximal CP violation.
