Christian Hennig's contribution to the Discussion of ‘Assumption-Lean Inference for Generalised Linear Model Parameters’ by Vansteelandt and Dukes

原文摘要

There is a tendency in statistics to talk about model assumptions in a misleading way. Most of us probably agree with George Box's ‘all models are wrong but some are useful’, yet there is much communication that implies that for applying methods ‘assuming’ certain models, these models have to be true. If this were so, no model-based method could ever be used! Generally model assumptions do not have to be fulfilled. A model assumption just means that certain theoretical results regarding a statistical procedure hold assuming the model. A procedure may well deliver useful results if its model assumptions do not hold. This can be addressed by investigating what happens if other models hold. The advantage of ‘assumption-lean’ methods is that they come with theory that applies under a wider range of models, so we know more, but none of this wider range of models will ultimately be ‘correct’ either, and the theory does not necessarily guarantee a good behaviour in practice. The authors shift the focus on estimands rather than models, although their theory and even their very definition still assumes a model, be it ‘big’ and non-parametric. Tukey (1997) rather put a focus on procedures, to be challenged by studying their behaviour under different models, without assuming any of them to be true. An important issue not treated by Tukey is addressed by discussing estimands, namely whether what we estimate is what we really want to know. This requires arguments other than evidence from the data. It may not be enough to choose an estimand so that its estimator fulfils nice asymptotic theory. The expected value E is central for the authors' estimands. Concentrating on one-dimensional location, standard robustness theory (Huber & Ronchetti, 2009) teaches that E is a non-robust functional. Looking at a model such as Pϵ,μ1,σ12,μ2,σ22=(1-ϵ)N(μ1,σ12)+ϵN(μ2,σ22) with very small ϵ⁠, theoretically all is fine with estimating E, but N(μ2,σ22) may model irregular outliers, and the interest maybe in estimating µ 1. But EPϵ,μ1,σ12,μ2,σ22 can be arbitrarily far away from µ1. The influence function of E is unbounded (as is (13) in the paper, due to the key role of E in the estimand), which is a problem from the robustness point of view. Paraphrasing Buja et al. (2019), having a general nonparametric theory for an estimand does not necessarily imply that the estimand behaves well over the range of models covered by the theory. Whether an estimand is well chosen requires more than being ‘assumption-lean’.

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