  The smallness of the $\theta_{13}$ mixing angle as observed in neutrino
oscillation experiments can be understood through an approximated $\mu - \tau$
exchange symmetry in the neutrino mass matrix. Using recent oscillation
neutrino data, but assuming no \textit{CP} violation, we study $\mu-\tau$
breaking parameter space to establish the conditions under which such a
breaking could have a perturbative origin. According to the so-obtained
conditions, we suggest that a sterile neutrino, matching LSND/MiniBooNE
neutrino oscillation results, could provide the necessary ingredients to
properly fix atmospheric and $\theta_{13}$ mixing angles to observable values,
without exceeding the sterile neutrino fraction bound in solar oscillations. In
such a scenario, we analyze the general effect of a fourth neutrino on the
prediction for the effective $m_{ee}$ majorana mass parameter.
