  We study an inflationary scenario with a two-form field to which an inflaton
couples non-trivially. First, we show that anisotropic inflation can be
realized as an attractor solution and that the two-form hair remains during
inflation. A statistical anisotropy can be developed because of a cumulative
anisotropic interaction induced by the background two-form field. The power
spectrum of curvature perturbations has a prolate-type anisotropy, in contrast
to the vector models having an oblate-type anisotropy. We also evaluate the
bispectrum and trispectrum of curvature perturbations by employing the in-in
formalism based on the interacting Hamiltonians. We find that the non-linear
estimators $f_{NL}$ and $\tau_{NL}$ are correlated with the amplitude $g_*$ of
the statistical anisotropy in the power spectrum. Unlike the vector models,
both $f_{NL}$ and $\tau_{NL}$ vanish in the squeezed limit. However, the
estimator $f_{NL}$ can reach the order of 10 in the equilateral and enfolded
limits. These results are consistent with the latest bounds on $f_{NL}$
constrained by Planck.
