Search for: All records

Is there a natural way to order data in dimension greater than one? The approach based on the notion of data depth, often associated with the name of John Tukey, is among the most popular. Tukey's depth has found applications in robust statistics, graph theory, and the study of elections and social choice. We present improved performance guarantees for empirical Tukey's median, a deepest point associated with the given sample, when the data-generating distribution is elliptically symmetric and possibly anisotropic. Some of our results remain valid in the wider class of affine equivariant estimators. As a corollary of our bounds, we show that the typical diameter of the set of all empirical Tukey's medians scales like $$o(n^{-1/2})$$ where $$n$$ is the sample size. Moreover, when the data are 2-dimensional, we prove that with high probability, the diameter is of order $$O(n^{-3/4}\log^{3/2}(n))$$. 

Abstract Droplet‐based single cell sequencing technologies, such as inDrop, Drop‐seq, and 10X Genomics, are catalyzing a revolution in the understanding of biology. Barcoding beads are key components for these technologies. What is limiting today are barcoding beads that are easy to fabricate, can efficiently deliver primers into drops, and thus achieve high detection efficiency. Here, this work reports an approach to fabricate dissolvable polyacrylamide beads, by crosslinking acrylamide with disulfide bridges that can be cleaved with dithiothreitol. The beads can be rapidly dissolved in drops and release DNA barcode primers. The dissolvable beads are easy to synthesize, and the primer cost for the beads is significantly lower than that for the previous barcoding beads. Furthermore, the dissolvable beads can be loaded into drops with >95% loading efficiency of a single bead per drop and the dissolution of beads does not influence reverse transcription or the polymerase chain reaction (PCR) in drops. Based on this approach, the dissolvable beads are used for single cell RNA and protein analysis.