Robustness of Consistent Loss Functions for Multinomial Outcome Models

Abstract

Maximum likelihood estimation, which uses the logarithmic loss function, is the default method used to estimate latent parameters consistently in multinomial outcome models. However, it is sensitive to even a tiny fraction of corruption in the training data. Alternatively, other loss functions in the family of strictly consistent loss functions can be used to consistently estimate model parameters. In this thesis, we study the robustness properties of different loss functions in the family, mainly the logarithmic loss function, the quadratic loss function, and the spherical loss function. We introduce two notions of robustness properties of loss functions. A loss function is partially robust if its corresponding influence function, a proxy for the bias from corruption, has bounded 2-norm. On the other hand, a loss function is strongly robust if the 2-norm of the bias itself is bounded. When some mild assumptions are met, the quadratic loss function can be shown to be both partially robust and strongly robust, while the logarithmic loss function is not. We also demonstrate that the behaviors of each loss function agree with their theoretical properties when used to estimate parameter in two synthetic models: a price-purchase model and a multinomial logit with intercepts model for two products. This thesis thus not only advocates more use of the quadratic loss function in parameter estimation of multinomial outcome models but also serves as a framework to conduct future research of the cross section between the robustness of loss functions and the consistency of parameter estimation.