In this paper, we consider a minimax approach to the estimation and filtering problems in the stochastic framework, where covariances of the random factors are completely unknown. The term 'random factors' refers either to unknown parameters and measurement noise in the estimation problem or to disturbance process and the initial state of a linear discrete-time dynamic system in the filtering problem. We introduce a notion of the attenuation level of random factors as a performance measure for both a linear unbiased estimate and a filter. This is the worst-case variance of the estimation error normalised by the sum of variances of all random factors over all nonzero covariance matrices. It is shown that this performance measure is equal to the spectral norm of the 'transfer matrix' and therefore the minimax estimate and filter can be computed in terms of linear matrix inequalities (LMIs). Moreover, the explicit formulae for both the minimax estimate and the minimal value of the attenuation level are presented in the estimation problem. It turns out that the above attenuation level of random factors coincides with the attenuation level of deterministic factors that is the worst-case normalised squared Euclidian norm of the estimation error over all nonzero sample values of random factors. In addition, we demonstrate that the LMI technique can be applied to derive the optimal robust estimator and filter, when there is a priori information about convex polyhedral sets which unknown covariance matrices of random factors belong to. Two illustrative examples show advantages of the minimax approach proposed.