Chaohua Dong, Jiti Gao and Oliver Linton’s Contribution to the Discussion of ‘Assumption-Lean Inference for Generalised Linear Model Parameters’ by Vansteelandt and Dukes
本文是一篇讨论性文章，批评了原论文在广义线性模型参数推断中缺乏有意义假设的问题，指出其估计量并非半参数有效，且对高维协变量的处理存在维度困境。
The title of this paper is ironically self-fulfilling, since there are almost no meaningful assumptions made throughout! The starting point is that we have some plausible semi-parametric model, which is a special case of a more general non-parametric model, but we wish to allow for misspecification and in particular define an estimand that is meaningful in the non-parametric model and that specialises in the plausible model to a slope coefficient. However, since the estimator that is proposed is not the semi-parametric efficient estimator of that slope coefficient under the semi-parametric model, we wonder what is the role of the model at all? The theory side of it seems to assume in Theorem 2, for example that E(Y | A, L) is consistently estimated in L 2 under the full unrestricted -parametric setting. But if that is possible, then why bother with the model? The authors talk casually about machine learning methods being used to estimate E(Y | A, L), but if that is a silver bullet, then who needs the model? The model embodies some structure around A, L but the discussion is focussed away from the dimensionality of L, which is a big reason why one might want a structured model such as additivity Perhaps it would help if a full model was written down for the effect of high-dimensional L. Perhaps the point is that the parameter of interest is only defined in terms of low-dimensional conditional expectations, but this does not appear to be the case in the sense that high-dimensional smoothing is employed in (a) of p.14, which is then projected down by conditional expectation onto L, but if A is binary, then this has not reduced dimensionality at all, the dimensionality issue sits in L and what structure is assumed about its effect on Y.