A general class of discrete-time uncertain nonlinear stochastic systems with quadratic sum constraints is considered. A linear fixed order state estimator for state estimation is presented for various estimation error performance criteria in a unified framework. The observer is of order equal to the difference between the state and output vector dimensions. The performance criteria considered in this paper include guaranteed-cost suboptimal versions of estimation objectives like H₂, H_∞, stochastic passivity, etc. The design of fixed-order linear state estimators that satisfy these criteria are given using a common matrix inequality formulation.