A minimax filtering problem for a class of uncertain linear stochastic systems is studied. Uncertainties involving fourth-order moments of the noise distribution and of the initial state are considered. Under the hypothesis that the second-order statistics are known (hence the linear filter is available) the quadratic minimax filter is found. Moreover, it is shown that the minimax filter gives a worst case error variance even less than the exact error variance of the linear filter. Numerical simulations show that this improvement is meaningful also in cases of "great uncertainty" regarding the higher order statistics.