  Cosmic rays are the highest energy particles available for our study and as
such serve as excellent probes of the effects of Lorentz Invariance Violations,
which are expected to increase with energy. This general paradigm is
investigated in this paper by studying the effects of such violations within
the Coleman-Glashow model in which each particle species may have its own
maximum attainable velocity, even exceeding that of light \textit{in vacuo}.
The particular focus here is that the muon neutrino may have the maximum speed
exceeding that of light. We show that such an assumption leads to the
elongation of the decay lifetime of the pion that increases with energy over
and above the time dilation effects. We provide a transparent analytical
derivation of the spectral intensities of muon neutrinos and muons generated in
the Earth's atmosphere by cosmic rays. In this derivation we not only account
for elongation of the pion lifetime, but also for the loss of energy by the
neutrinos by radiation of the electron-positron pairs through the Cohen-Glashow
process, during their propagation. We then compare the theoretical spectra with
observations of neutrinos and muons from large instruments like IceCube and
BUST to set a limit of $\sim10^{-13}$ on the fractional excess speed of
neutrinos over that of light. We also show that the ratio of the spectral
intensities of downward and upward moving neutrinos at various angles
constitute a diagnostic exclusively for the Cohen-Glashow process, which may be
searched for in the IceCube data set. We conclude the paper with several
comments, including those related to improvements of these tests when definite
signals of GZK neutrinos will be observed.
