This paper considers a robust filtering problem for a linear discrete time invariant system with measured and estimated outputs. The system is exposed to random disturbances with imprecisely known dis tributions generated by an unknown stable shaping filter from the Gaussian white noise. The stochastic uncertainty of the input disturbance is measured by the mean anisotropy functional. The estimation error is quantified by the anisotropic norm which is a stochastic analogue of the H∞ norm. A sufficient condition for an estimator to exist and ensure that the error is less than a given threshold value is derived in form of a convex inequality on the determinant of a positive definite matrix and two linear matrix inequalities. The suboptimal problem setting results to a set of the estimators ensuring the anisotropic norm of the error to be strictly bounded thereby providing some additional degree of freedom to impose some additional constraints on the estimator performance specification.