Communications in Mathematics and Statistics最新文献

For any finitely generated unital commutative associative algebra (mathcal {R}) over (mathbb {C}) and any complex finite-dimensional simple Lie algebra (mathfrak {g}) with a fixed Cartan subalgebra (mathfrak {h}), we classify all (mathfrak {g}otimes mathcal {R})-modules on (U(mathfrak {h})) such that (mathfrak {h}) as a subalgebra of (mathfrak {g}otimes mathcal {R}), acts on (U(mathfrak {h})) by the multiplication. We construct these modules explicitly and study their module structures.

Abstract

Causal inference and missing data have attracted significant research interests in recent years, while the current literature usually focuses on only one of these two issues. In this paper, we develop two multiply robust methods to estimate the quantile treatment effect (QTE), in the context of missing data. Compared to the commonly used average treatment effect, QTE provides a more complete picture of the difference between the treatment and control groups. The first one is based on inverse probability weighting, the resulting QTE estimator is root-n consistent and asymptotic normal, as long as the class of candidate models of propensity scores contains the correct model and so does that for the probability of being observed. The second one is based on augmented inverse probability weighting, which further relaxes the restriction on the probability of being observed. Simulation studies are conducted to investigate the performance of the proposed method, and the motivated CHARLS data are analyzed, exhibiting different treatment effects at various quantile levels.

Discretization of continuous-time diffusion processes is a widely recognized method for sampling. However, the canonical Euler Maruyama discretization of the Langevin diffusion process, referred as unadjusted Langevin algorithm (ULA), studied mostly in the context of smooth (gradient Lipschitz) and strongly log-concave densities, is a considerable hindrance for its deployment in many sciences, including statistics and machine learning. In this paper, we establish several theoretical contributions to the literature on such sampling methods for non-convex distributions. Particularly, we introduce a new mixture weakly smooth condition, under which we prove that ULA will converge with additional log-Sobolev inequality. We also show that ULA for smoothing potential will converge in (L_{2})-Wasserstein distance. Moreover, using convexification of nonconvex domain (Ma et al. in Proc Natl Acad Sci 116(42):20881–20885, 2019) in combination with regularization, we establish the convergence in Kullback–Leibler divergence with the number of iterations to reach (epsilon )-neighborhood of a target distribution in only polynomial dependence on the dimension. We relax the conditions of Vempala and Wibisono (Advances in Neural Information Processing Systems, 2019) and prove convergence guarantees under isoperimetry, and non-strongly convex at infinity.

This article is focused on the problem to measure and test the conditional mean dependence of a response variable on a predictor variable. A local influence detection approach is developed combining with the martingale difference divergence (MDD) metric, and an efficient wild bootstrap implementation is given. The obtained new metric of the conditional mean dependence holds the merits of MDD, while it is more sensitive than the original one, and leads to a powerful test to nonlinear relationships. It is shown by simulations that the proposed test can achieve higher power for general conditional mean dependence relationships even in high-dimensional settings. Theoretical asymptotic properties of the local influence test statistic are given, and a real data analysis is also presented for further illustration. The localization idea could be combined with other conditional mean dependence metrics.