Robust Inference and Modeling of Mean and Dispersion for Generalized Linear Models

Abstract

Generalized Linear Models (GLMs) are a popular class of regression models when the responses follow a distribution in the exponential family. In real data the variability often deviates from the relation imposed by the exponential family distribution, which results in over- or underdispersion. Dispersion effects may even vary in the data. Such datasets do not follow the traditional GLM distributional assumptions, leading to unreliable inference. Therefore, the family of double exponential distributions has been proposed, which models both the mean and the dispersion as a function of covariates in the GLM framework. Since standard maximum likelihood inference is highly susceptible to the possible presence of outliers, we propose the robust double exponential (RDE) estimator. Asymptotic properties and robustness of the RDE estimator are discussed. A generalized robust quasi-deviance measure is introduced which constitutes the basis for a stable robust test. Simulations for binomial and Poisson models show the excellent performance of the RDE estimator and corresponding robust tests. Penalized versions of the RDE estimator are developed for sparse estimation with high-dimensional data and for flexible estimation via generalized additive models (GAMs). Real data applications illustrate the relevance of robust inference for dispersion effects in GLMs and GAMs. Supplementary materials for this article are available online.

KEYWORDS:

• Double exponential family
• Influence function
• Likelihood ratio test
• M-estimator
• Penalization

Supplementary Materials

The supplementary material provides expressions for the influence function and variance of the robust double exponential (RDE) estimator. It also contains additional simulation results to investigate the performance of the RDE estimator, its corresponding tests and the GAM and sparse RDE estimators. An extra example illustrates RDE inference in a double binomial model. Moreover, the assumptions for the theoretical results of the robust test based on RDE are stated, an outline for the proof of Corollary 1 is provided and a high-dimensional asymptotic analysis establishes consistency and oracle properties of sparse RDE estimators under appropriate assumptions.

Additional information

Funding

The authors gratefully acknowledge the support of International Funds KU Leuven, grant C16/15/068 and the BASF Research Chair on Robust Predictive Analytics at KU Leuven.