  We prove decorrelation estimates for generalized lattice Anderson models on
$Z^d$ constructed with finite-rank perturbations in the spirit of Klopp
\cite{klopp}. These are applied to prove that the local eigenvalue statistics
$\xi^\omega_{E}$ and $\xi^\omega_{E^\prime}$, associated with two energies $E$
and $E'$ satisfying $|E - E'| > 4d$, are independent. That is, if $I,J$ are two
bounded intervals, the random variables $\xi^\omega_{E}(I)$ and
$\xi^\omega_{E'}(J)$, are independent and distributed according to a compound
Poisson distribution whose L\'evy measure has finite support. We also prove
that the extended Minami estimate implies that the eigenvalues in the
localization region have multiplicity at most the rank of the perturbation.
