We prove decorrelation estimates for generalized lattice Anderson models on Zd constructed with finite-rank perturbations in the spirit of Klopp . These are applied to prove that the local eigenvalue statistics ! ωE and ! ωE0 , associated with two energies E and E0 in the localization region and satisfying jE E0j > 4d, are independent. That is, if I; J are two bounded intervals, the random variables ! ωE.I / and ! ωE0.J /, are independent and distributed according to a compound Poisson distribution whose Levy 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 [12, 19, 21], extends tomodels forwhich the eigenvalues are degenerate, and applies tomodels forwhich the potential is not sign definite  in dimensions d > 1. © 2021 European Mathematical Society.