We are interested in the problem of the estimation of states and some unknown parameters for a class of discrete-time nonlinear state-space model. The class contains those nonlinear models written using a Linear Parameter Varying (LPV) or a quasi-LPV form. The unknown parameters appear affinely in the LPV/quasi-LPV form. An elimination approach forms the core of the proposed algorithm that leads to the estimation of the parameter and then the states in a sliding measurement window. Several examples illustrate the mechanics and the working of the algorithm. Some insights and implications of this approach are discussed.