Estimating GEV models with censored data

We examine the problem of estimating parameters for Generalized Extreme Value (GEV) models when one or more alternatives are censored in the sample data, i.e., all decision makers who choose these censored alternatives are excluded from the sample; however, information about the censored alternatives is still available. This problem is common in marketing and revenue management applications, and is essentially an extreme form of choice-based sampling. We review estimators typically used with GEV models, describe why many of these estimators cannot be used for these censored samples, and present two approaches that can be used to estimate parameters associated with censored alternatives. We detail necessary conditions for the identification of parameters associated exclusively with the utility of censored alternatives. These conditions are derived for single-level nested logit, multi-level nested logit and cross-nested logit models. One of the more surprising results shows that alternative specific constants for multiple censored alternatives that belong to the same nest can still be separately identified in nested logit models. Empirical examples based on simulated datasets demonstrate the large-sample consistency of estimators and provide insights into data requirements needed to estimate these models for finite samples.