On the heavy-tail behavior of the distributionally robust newsvendor

Since the seminal work of Scarf (1958) [A min-max solution of an inventory problem, Studies in the Mathematical Theory of Inventory and Production, pages 201-209] on the newsvendor problem with ambiguity in the demand distribution, there has been a growing interest in the study of the distributionally robust newsvendor problem. The optimal order quantity is computed by accounting for the worst possible distribution from a set of demand distributions that is characterized by partial information, such as moments. The model is criticized at times for being overly conservative since the worst-case distribution is discrete with a few support points. However, it is the order quantity from the model that is typically of practical relevance. A simple observation shows that the optimal order quantity in Scarf's model with known first and second moment is also optimal for a heavy-tailed censored student-t distribution with degrees of freedom 2. In this paper, we generalize this "heavy-tail optimality" property of the distributionally robust newsvendor to a more general ambiguity set where information on the first and the $n$th moment is known, for any real number $n > 1$. We provide a characterization of the optimal order quantity under this ambiguity set by showing that for high critical ratios, the order quantity is optimal for a regularly varying distribution with an approximate power law tail with tail index $n$. We illustrate the applicability of the model by calibrating the ambiguity set from data and comparing the performance of the order quantities computed via various methods in a dataset.