Use of uncertain additional information in newsvendor models

The newsvendor problem is a popular inventory management problem in supply chain management and logistics. Solutions to the newsvendor problem determine optimal inventory levels. This model is typically fully determined by a purchase and sale prices and a distribution of random market demand. From a statistical point of view, this problem is often considered as a quantile estimation of a critical fractile which maximizes anticipated profit. The distribution of demand is a random variable and is often estimated on historic data. In an ideal situation, when the probability distribution of demand is known, one can determine the quantile of a critical fractile minimizing a particular loss function. When a parametric family is known, maximum likelihood estimation is asymptotically efficient under certain regularity assumptions and the maximum likelihood estimators (MLEs) are used for estimating quantiles. Then, the Cramer-Rao lower bound determines the lowest possible asymptotic variance for the MLEs. Can one find a quantile estimator with a smaller variance then the Cramer-Rao lower bound? If a relevant additional information is available then the answer is yes. This manuscript considers minimum variance and mean squared error estimation which incorporate additional information for estimating optimal inventory levels