In this paper, we study assortment planning under the marginal distribution model (MDM), a semiparametric choice model that only requires information about the marginal noise in the utilities of alternatives and does not assume independence of the noise terms. It is already known in the literature that the multinomial logit (MNL) model belongs to the MDM framework. In this work, we demonstrate that some multi-purchase choice models such as the multiple-discrete-choice (MDC) model and the threshold utility model (TUM) also fall into the framework of MDM, even though MDM does not explicitly model multi-purchase behavior. For assortment problems within the MDM framework, we identify a general condition under which a strictly profit-nested assortment is optimal. While the problem is NP-hard, we show that the best strictly profit-nested assortment is a 1/2-approximate solution for all MDMs. Additionally, we present a simple example of an MDM for which the 1/2-approximate bound is tight. These results either extend or improve upon previous findings on assortment optimization under MNL, MDC, and TUM. Additionally, we present an arbitrary-close approximation algorithm for MDM, and an improved version for a class of choice models that includes MDC as a special case. Finally, we conduct experiments on real-world data and compare the predictive power of several choice models in the presence of multi-purchase behavior.