  We prove some local estimates on the trace of spectral projectors for random
Schr\"odinger operators restricted to cubes $\Lambda \subset R^d$. We also
present a new proof of the spectral averaging result based on analytic
perturbation theory. Together, these provide another proof of the Wegner
estimate with an explicit form of the constant and an alternate proof of the
Birman-Solomyak formula. We also use these results to prove the Lipschitz
continuity of the local density of states function for a restricted family of
random Schr\"odinger operators on cubes $\Lambda \subset R^d$, for $d \geq 1$.
The result holds for low energies without a localization assumption but is not
strong enough to extend to the infinite-volume limit.
