Demand Modeling in the Presence of Unobserved Lost Sales

We present an integrated optimization approach to parameter estimation for discrete choice demand models where data for one or more choice alternatives are censored. We employ a mixed-integer program (MIP) to jointly determine the prediction parameters associated with the customer arrival rate and their substitutive choices. This integrated approach enables us to recover proven, (near-) global optimal parameter values with respect to the chosen loss-minimization objective function, thereby overcoming a limitation of prior approaches that employ multi-start heuristic procedures and terminate without providing precise information on the solution quality. The imputations are done endogenously in the MIP by estimating optimal values for the probabilities of the unobserved choices being selected. Under mild assumptions, we prove that the approach is asymptotically consistent. Partial information, user acceptance criteria, model selection, and regularization techniques can be incorporated to enhance practical efficacy. We test the method on simulated and real data, and present results for a variety of singleand multi-item demand prediction scenarios, and for learning the unobserved market shares of competitors.