Centers and Azumaya loci of finite WW-algebras

In this paper, we study the center $Z$ of the finite $W$-algebra $\mathcal{T}(\mathfrak{g},e)$ associated with a semi-simple Lie algebra $\mathfrak{g}$ over an algebraically closed field \mathds{k} of characteristic $p\gg0$, and an arbitrarily given nilpotent element $e\in\mathfrak{g}$. We obtain an analogue of Veldkamp's theorem on the center. For the maximal spectrum $\text{Specm}(Z)$, we show that its Azumaya locus coincides with its smooth locus of smooth points. The former locus reflects irreducible representations of maximal dimension for $\mathcal{T}(\mathfrak{g},e)$.Comment: 31 page

Combining multiple observational data sources to estimate causal effects

The era of big data has witnessed an increasing availability of multiple data sources for statistical analyses. We consider estimation of causal effects combining big main data with unmeasured confounders and smaller validation data with supplementary information on these confounders. Under the unconfoundedness assumption with completely observed confounders, the smaller validation data allow for constructing consistent estimators for causal effects, but the big main data can only give error-prone estimators in general. However, by leveraging the information in the big main data in a principled way, we can improve the estimation efficiencies yet preserve the consistencies of the initial estimators based solely on the validation data. Our framework applies to asymptotically normal estimators, including the commonly-used regression imputation, weighting, and matching estimators, and does not require a correct specification of the model relating the unmeasured confounders to the observed variables. We also propose appropriate bootstrap procedures, which makes our method straightforward to implement using software routines for existing estimators

Asymptotic causal inference with observational studies trimmed by the estimated propensity scores

Causal inference with observational studies often relies on the assumptions of unconfoundedness and overlap of covariate distributions in different treatment groups. The overlap assumption is violated when some units have propensity scores close to 0 or 1, and therefore both practical and theoretical researchers suggest dropping units with extreme estimated propensity scores. However, existing trimming methods ignore the uncertainty in this design stage and restrict inference only to the trimmed sample, due to the non-smoothness of the trimming. We propose a smooth weighting, which approximates the existing sample trimming but has better asymptotic properties. An advantage of the new smoothly weighted estimator is its asymptotic linearity, which ensures that the bootstrap can be used to make inference for the target population, incorporating uncertainty arising from both the design and analysis stages. We also extend the theory to the average treatment effect on the treated, suggesting trimming samples with estimated propensity scores close to 1.Comment: 21 pages, 1 figures and 3 table