  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 $E'$ in the localization region and 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. The method of proof contains new ingredients that simplify
the proof of the rank one case \cite{klopp,shirley,trinh}, extends to models
for which the eigenvalues are degenerate, and applies to models for which the
potential is not sign definite \cite{tautenhahn-veselic1} in dimensions $d \geq
1$.
