  We analyze the neutrino events from the supernova (SN) 1987A detected by the
Kamiokande II (KII) and Irvine-Michigan-Brookhaven (IMB) experiments. For the
time-integrated flux we assume a quasi-thermal spectrum of the form
$(E/E_0)^\alpha e^{-(\alpha+1)E/E_0}$ where $\alpha$ plays the role of a
spectral index. This simple representation not only allows one to fit the total
energy $E_{\rm tot}$ emitted in $\bar\nu_e$ and the average energy
$<E_{\bar\nu_e}>$, but also accommodates a wide range of shapes, notably
anti-pinched spectra that are broader than a thermal distribution. We find that
the pile-up of low-energy events near threshold in KII forces the best-fit
value for $\alpha$ to the lowest value of any assumed prior range. This applies
to the KII events alone as well as to a common analysis of the two data sets.
The preference of the data for an ``unphysical'' spectral shape implies that
one can extract meaningful values for $<E_{\bar\nu_e}>$ and $E_{\rm tot}$ only
if one fixes a prior value for $\alpha$. The tension between the KII and IMB
data sets and theoretical expectations for $<E_{\bar\nu_e}>$ is not resolved by
an anti-pinched spectrum.
