  We examine the MiniBooNE neutrino, MiniBooNE antineutrino and LSND
antineutrino data sets in a two-neutrino
$\stackrel{\tiny{(-)}}{\nu}_{\mu}\rightarrow\stackrel{\tiny{(-)}}{\nu}_e$
oscillation approximation subject to non-standard matter effects. We assume
those effects can be parametrized by an $L$-independent effective potential,
$V_s=\pm A_s$, experienced only by an intermediate, non-weakly-interacting
(sterile) neutrino state which we assume participates in the oscillation, where
$+/-$ corresponds to neutrino/antineutrino propagation. We discuss the
mathematical framework in which such oscillations arise in detail, and derive
the relevant oscillation probability as a function of the vacuum oscillation
parameters $\Delta m^2$ and $\sin^22\theta_{\mu e}$, and the matter effect
parameter $A_s$. We are able to successfully fit all three data sets, including
the MiniBooNE low energy excess, with the following best-fit model parameters:
$\Delta m^2=0.47$ eV$^2$, $\sin^22\theta_{\mu e}=0.010$, and
$A_s=2.0\times10^{-10}$ eV. The $\chi^2$-probability for the best fit
corresponds to 21.6%, to be compared to 6.8% for a fit where $A_s$ has been set
to zero, corresponding to a (3+1) sterile neutrino oscillation model. We find
that the compatibility between the three data sets corresponds to 17.4%, to be
compared to 2.3% for $A_s=0$. Finally, given the fit results, we examine
consequences for reactor, solar, and atmospheric oscillations. For this paper,
the presented model is empirically driven, but the results obtained can be
directly used to investigate various phenomenological interpretations such as
non-standard matter effects.
