In this paper, we consider the design of robust linear precoders for MU-MISO systems where users have perfect Channel State Information (CSI) while the BS has partial CSI. In particular, the BS has access to imperfect estimates of the channel vectors, in addition to the covariance matrices of the estimation error vectors. A closed-form expression for the Average Minimum Mean Square Error (AMMSE) is obtained using the second order Taylor Expansion. This approximation is used to formulate two fairness-based robust design problems: a maximum AMMSE-constrained problem and a power-constrained problem. We propose an algorithm based on convex optimization techniques to address the first problem, while the second problem is tackled by exploiting the close relationship between the two problems, in addition to their monotonie natures.