Honest adaptive confidence bands and self-similar functions

Confidence bands are confidence sets for an unknown function f, containing all functions within some sup-norm distance of an estimator. In the density estimation, regression, and white noise models, we consider the problem of constructing adaptive confidence bands, whose width contracts at an optimal rate over a range of H\"older classes. While adaptive estimators exist, in general adaptive confidence bands do not, and to proceed we must place further conditions on f. We discuss previous approaches to this issue, and show it is necessary to restrict f to fundamentally smaller classes of functions. We then consider the self-similar functions, whose H\"older norm is similar at large and small scales. We show that such functions may be considered typical functions of a given H\"older class, and that the assumption of self-similarity is both necessary and sufficient for the construction of adaptive bands. Finally, we show that this assumption allows us to resolve the problem of undersmoothing, creating bands which are honest simultaneously for functions of any H\"older norm

Spatially-adaptive sensing in nonparametric regression

While adaptive sensing has provided improved rates of convergence in sparse regression and classification, results in nonparametric regression have so far been restricted to quite specific classes of functions. In this paper, we describe an adaptive-sensing algorithm which is applicable to general nonparametric-regression problems. The algorithm is spatially adaptive, and achieves improved rates of convergence over spatially inhomogeneous functions. Over standard function classes, it likewise retains the spatial adaptivity properties of a uniform design

Near-optimal estimation of jump activity in semimartingales

In quantitative finance, we often model asset prices as semimartingales, with drift, diffusion and jump components. The jump activity index measures the strength of the jumps at high frequencies, and is of interest both in model selection and fitting, and in volatility estimation. In this paper, we give a novel estimate of the jump activity, together with corresponding confidence intervals. Our estimate improves upon previous work, achieving near-optimal rates of convergence, and good finite-sample performance in Monte-Carlo experiments.The author acknowledges the EPSRC for their support under grant EP/K000993/1.This is the final version of the article. It was first available from Institute of Mathematical Statistics via http://dx.doi.org/10.1214/15-AOS134

A Smirnov-Bickel-Rosenblatt theorem for compactly-supported wavelets

In nonparametric statistical problems, we wish to find an estimator of an unknown function f. We can split its error into bias and variance terms; Smirnov, Bickel and Rosenblatt have shown that, for a histogram or kernel estimate, the supremum norm of the variance term is asymptotically distributed as a Gumbel random variable. In the following, we prove a version of this result for estimators using compactly-supported wavelets, a popular tool in nonparametric statistics. Our result relies on an assumption on the nature of the wavelet, which must be verified by provably-good numerical approximations. We verify our assumption for Daubechies wavelets and symlets, with N = 6, ..., 20 vanishing moments; larger values of N, and other wavelet bases, are easily checked, and we conjecture that our assumption holds also in those cases

Long-wavelength TCF-based fluorescence probes for the detection and intracellular imaging of biological thiols

Two ‘turn on’ TCF-based fluorescence probes were developed for the detection of biological thiols (TCF-GSH and TCFCl-GSH). TCF-GSH was shown to have a high sensitivity towards glutathione (GSH) with a 0.28 μM limit of detection. Unfortunately, at higher GSH concentrations the fluorescence intensity of TCF-GSH decreased and toxicity was observed for TCF-GSH in live cells. However, TCFCl-GSH was shown to be able to detect GSH at biologically relevant concentrations with a 0.45 μM limit of detection. No toxicity was found for TCFCl-GSH and a clear ‘turn on’ with good photostability was observed for the exogenous addition of GSH, Cys and HCys. Furthermore, TCFCl-GSH was used to evaluate the effects of drug treatment on the levels of GSH in live cells

A simple umbelliferone based fluorescent probe for the detection of nitroreductase

A simple nitrobenzyl-umbelliferone (NCOU1) was synthesised containing a nitroreductase (NTR) trigger moiety. The presence of NTR, resulted in the fragmentation of the parent molecule and release of the highly emissive fluorophore umbelliferone via an NTR-catalyzed reduction of the nitro group. In the presence of the NTR enzyme, NCOU1 gave rise to a 5-fold increase in fluorescence intensity at 455 nm and was selective for NTR over other reductive enzymes. These results indicate that NCOU1 can be used as a simple assay for the detection of NTR

Polymer of Intrinsic Microporosity Induces Host-Guest Substrate Selectivity in Heterogeneous 4-Benzoyloxy-TEMPO-Catalysed Alcohol Oxidations

The free radical 4-benzoyloxy-2,2,6,6-tetramethylpiperidine-1-oxyl (4B-TEMPO) is active as an electrocatalyst for primary alcohol oxidations when immobilised at an electrode surface and immersed into an aqueous carbonate buffer solution. In order to improve the catalytic process, a composite film electrode is developed based on (i) carbon microparticles of 2–12 μm diameter to enhance charge transport and (ii) a polymer of intrinsic microporosity (here PIM-EA-TB with a BET surface area of 1027 m2 g−1). The latter acts as a highly rigid molecular framework for the embedded free radical catalyst with simultaneous access to aqueous phase and substrate. The resulting mechanism for the oxidation of primary alcohols is shown to switch in reaction order from first to zeroth with increasing substrate concentration consistent with a kinetically limited process with competing diffusion of charge at the polymer layer-electrode interface (here the “LEk” case in Albery-Hillman notation). Reactivity optimisation and screening for a wider range of primary alcohols in conjunction with DFT-based relative reactivity correlation reveals substrate hydrophobicity as an important factor for enhancing catalytic currents. The PIM-EA-TB host matrix is proposed to control substrate partitioning and thereby catalyst reactivity and selectivity