Heavy-tailed errors impair the accuracy of the least squares estimate, which can be spoiled by a single grossly outlying observation. As argued in the seminal work of Peter Huber in 1973 [Ann. Statist. 1 (1973) 799–821], robust alternatives to the method of least squares are sorely needed. To achieve robustness against heavy-tailed sampling distributions, we revisit the Huber estimator from a new perspective by letting the tuning parameter involved diverge with the sample size. In this paper, we develop nonasymptotic concentration results for such an adaptive Huber estimator, namely, the Huber estimator with the tuning parameter adapted to sample size, dimension, and the variance of the noise. Specifically, we obtain a sub-Gaussian-type deviation inequality and a nonasymptotic Bahadur representation when noise variables only have finite second moments. The nonasymptotic results further yield two conventional normal approximation results that are of independent interest, the Berry-Esseen inequality and Cramér-type moderate deviation. As an important application to large-scale simultaneous inference, we apply these robust normal approximation results to analyze a dependence-adjusted multiple testing procedure for moderately heavy-tailed data. It is shown that the robust dependence-adjusted procedure asymptotically controls the overall false discovery proportion at the nominal level under mild moment conditions. Thorough numerical results on both simulated and real datasets are also provided to back up our theory.

Due to current data collection technology, our ability to gather data has surpassed our ability to analyze it. In particular, k-means, one of the simplest and fastest clustering algorithms, is ill-equipped to handle extremely large datasets on even the most powerful machines. Our new algorithm uses a sample from a dataset to decrease runtime by reducing the amount of data analyzed. We perform a simulation study to compare our sampling based k-means to the standard k-means algorithm by analyzing both the speed and accuracy of the two methods. Results show that our algorithm is significantly more efficient than the existing algorithm with comparable accuracy.

The purpose of this paper is to explore spatio-temporal pattern formation via invasion fronts in the one and two dimensional Keller-Segel chemotaxis model. In the one-dimensional case, simulations show that solutions that begin near an unstable equilibrium evolve into periodic patterns. These in turn evolve into new patterns through a process known as coarsening. In the two-dimensional case, we encounter only periodic patterns in the wake of the initial front. Transverse patterning only arises as a result of a transverse instability of these periodic patterns from the leading invasion front.

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