Sliced Inverse Regression (SIR) has been extensively used to reduce the dimension of the predictor space before performing regression. SIR is originally a model free method but it has been shown to actually correspond to the maximum likelihood of an inverse regression model with Gaussian errors. This intrinsic Gaussianity of standard SIR may explain its high sensitivity to outliers as observed in a number of studies. To improve robustness, the inverse regression formulation of SIR is therefore extended to non-Gaussian errors with heavy-tailed distributions. Considering Student distributed errors it is shown that the inverse regression remains tractable via an Expectation-Maximization (EM) algorithm. The algorithm is outlined and tested in presence of outliers both on simulated and real data, showing improved results in comparison to a number of other existing approaches.