E-grocery offers customers an alternative to traditional brick-and-mortar grocery retailing. Customers select e-grocery for convenience, making use of the home delivery at a selected time slot. In contrast to brick-and-mortar retailing, in e-grocery on-stock information for stock keeping units (SKUs) becomes transparent to the customer before substantial shopping effort has been invested, thus reducing the personal cost of switching to another supplier. As a consequence, compared to brick-and-mortar retailing, on-stock availability of SKUs has a strong impact on the customers order decision, resulting in higher strategic service level targets for the e-grocery retailer. To account for these high service level targets, we propose a suitable model for accurately predicting the extreme right tail of the demand distribution, rather than providing point forecasts of its mean. Specifically, we propose the application of distributional regression methods so-called Generalised Additive Models for Location, Scale and Shape (GAMLSS)to arrive at the cost-minimising solution according to the newsvendor model. As benchmark models we consider linear regression, quantile regression, and some popular methods from machine learning. The models are evaluated in a case study, where we compare their out-of-sample predictive performance with regard to the service level selected by the e-grocery retailer considered.