  The Welch Bound is a lower bound on the root mean square cross correlation
between $n$ unit-norm vectors $f_1,...,f_n$ in the $m$ dimensional space
($\mathbb{R} ^m$ or $\mathbb{C} ^m$), for $n\geq m$. Letting $F =
[f_1|...|f_n]$ denote the $m$-by-$n$ frame matrix, the Welch bound can be
viewed as a lower bound on the second moment of $F$, namely on the trace of the
squared Gram matrix $(F'F)^2$. We consider an erasure setting, in which a
reduced frame, composed of a random subset of Bernoulli selected vectors, is of
interest. We extend the Welch bound to this setting and present the {\em
erasure Welch bound} on the expected value of the Gram matrix of the reduced
frame. Interestingly, this bound generalizes to the $d$-th order moment of $F$.
We provide simple, explicit formulae for the generalized bound for $d=2,3,4$,
which is the sum of the $d$-th moment of Wachter's classical MANOVA
distribution and a vanishing term (as $n$ goes to infinity with $\frac{m}{n}$
held constant). The bound holds with equality if (and for $d = 4$ only if) $F$
is an Equiangular Tight Frame (ETF). Our results offer a novel perspective on
the superiority of ETFs over other frames in a variety of applications,
including spread spectrum communications, compressed sensing and analog coding.
