Remote sensing pushbroom-type imaging systems acquire entire columns of an image with a single detector. As a consequence, the miss-calibration of the detectors produces stripes on the image. In this context, this paper introduces a new self-calibration destriping method based on an affine response model for the detectors, called Statistical Affine Destriping (SAD). In contrast, some previous contributions were limited to a purely linear model, while many others only considered an additive structured noise model. It is based on the maximum a posteriori estimation of the gain and offset parameters attached to each detector given the observed image. Simple statistical prior assumptions are adopted: respectively, a Gaussian white noise model for the gains and offsets, and a first-order, homogeneous Markov model for the observed scene. Based on a simplification of the posterior likelihood, we propose a very efficient optimization scheme based on a constrained Majorize-Minimize principle, allowing us to process large dimension images. Moreover, simple empirical rules are given to tune the hyperparameters of the destriping method for high-resolution PLEIADES-type images. Compared to the performance of a destriping method limited to gain correction, we observe that the new version provides reliable results in a wider range of situations. We also extend the method in two directions. On the one hand, we consider that some detectors may be atypical, with very high or very low gains or offsets. On the other hand, we extend the method to multispectral image destriping.