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We propose a bootstrap-based robust high-confidence level upper bound (Robust H-CLUB) for assessing the risks of large portfolios. The proposed approach exploits rank-based and quantile-based estimators, and can be viewed as a robust extension of the H-CLUB procedure (Fan etal., 2015). Such an extension allows us to handle possibly misspecified models and heavy-tailed data, which are stylized features in financial returns. Under mixing conditions, we analyze the proposed approach and demonstrate its advantage over H-CLUB. We further provide thorough numerical results to back up the developed theory, and also apply the proposed method to analyze a stock market dataset.

Cryo-electron microscopy (cryo-EM) has recently emerged as a powerful tool for obtaining three-dimensional (3D) structures of biological macromolecules in native states. A minimum cryo-EM image data set for deriving a meaningful reconstruction is comprised of thousands of randomly orientated projections of identical particles photographed with a small number of electrons. The computation of 3D structure from 2D projections requires clustering, which aims to enhance the signal to noise ratio in each view by grouping similarly oriented images. Nevertheless, the prevailing clustering techniques are often compromised by three characteristics of cryo-EM data: high noise content, high dimensionality and large number of clusters. Moreover, since clustering requires registering images of similar orientation into the same pixel coordinates by 2D alignment, it is desired that the clustering algorithm can label misaligned images as outliers. Herein, we introduce a clustering algorithm γ-SUP to model the data with a q-Gaussian mixture and adopt the minimum γ-divergence for estimation, and then use a self-updating procedure to obtain the numerical solution. We apply γ-SUP to the cryo-EM images of two benchmark macromolecules, RNA polymerase II and ribosome. In the former case, simulated images were chosen to decouple clustering from alignment to demonstrate γ-SUP is more robust to misalignment outliers than the existing clustering methods used in the cryo-EM community. In the latter case, the clustering of real cryo-EM data by our γ-SUP method eliminates noise in many views to reveal true structure features of ribosome at the projection level.

Varying coefficient models are a useful extension of classical linear models. They arise naturally when one wishes to examine how regression coefficients change over different groups characterized by certain covariates such as age. The appeal of these models is that the coef. cient functions can easily be estimated via a simple local regression. This yields a simple one-step estimation procedure. We show that such a one-step method cannot be optimal when different coefficient functions admit different degrees of smoothness. This drawback can be repaired by using our proposed two-step estimation procedure. The asymptotic mean-squared error for the two-step procedure is obtained and is shown to achieve the optimal rate of convergence. A few simulation studies show that the gain by the two-step procedure can be quite substantial. The methodology is illustrated by an application to an environmental data set.

Nonparametric deconvolution problems require one to recover an unknown density when the data are contaminated with errors. Optimal global rates of convergence are found under the weighted <i>L</i>_<i>p</i>loss (1 <i>p</i> ). It appears that the optimal rates of convergence are extremely low for supersmooth error distributions. To resolve this difficulty, we examine how high the noise level can be for deconvolution to be feasible, and for the deconvolution estimate to be as good as the ordinary density estimate. It is shown that if the noise level is not too high, nonparametric Gaussian deconvolution can still be practical. Several simulation studies are also presented.