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In this paper, the robust finite-horizon filtering problem is investigated for a class of uncertain nonlinear discrete time-varying stochastic systems with multiple missing measurements and error variance constraints. The stochastic nonlinearities are described by statistical means which can cover several classes of well-studied nonlinearities. The measurement missing phenomenon is also considered. Sufficient conditions are derived for a finite-horizon filter to satisfy the estimation error variance constraints. These conditions are expressed in terms of the feasibility of a series of recursive linear matrix inequalities (RLMIs). An illustrative simulation example is given to show the the effectiveness of the proposed algorithm.